Download §5 Manifolds as topological spaces

Differentiable Manifolds. Autumn 2014
Theodore Voronov.
§5
Manifolds as topological spaces
Last updated: 23 October (5 November) 2014.
5.1
Manifold topology
Consider a manifold M n = (M, A), where A = (ϕ : Vα → Uα ) is a smooth atlas defining the
manifold structure on M . Here Vα ⊂ Rn and Uα ⊂ M .
Definition 5.1. We call a subset A ⊂ M open if and only if for each chart in the atlas A, the
n
set ϕ−1
α (A ∩ Uα ) ⊂ Vα ⊂ R is open.
Example 5.1. The codomain Uα of each chart is an open set. (We check this below, in the
proof of Theorem 5.2.)
Consider unions and intersections of open subsets of a manifold. Let ϕ : V → U be a
particular chart. (For the simplicity of notation we suppress the index α.) For an arbitrary
family of sets Aµ ⊂ M we have
[
[
[
ϕ−1 (( Aµ ) ∩ U ) = ϕ−1 ( (Aµ ∩ U )) =
ϕ−1 (Aµ ∩ U )),
\
\
\
ϕ−1 (( Aµ ) ∩ U ) = ϕ−1 ( (Aµ ∩ U )) =
ϕ−1 (Aµ ∩ U )).
because ϕ is a bijection. Also, ϕ−1 (∅) = ∅, so ∅ ⊂ M is open, and ϕ−1 (M ) = V is open.
We arrive at the following theorem1 :
Theorem 5.1. Open subsets of a manifold M (defined as above) satisfy the axioms of a topology.
Therefore each manifold can be considered as a topological space. We shall refer to the
topology defined above as to the manifold topology or the topology given by a manifold structure.
Note that so far we have defined topology of M n using one particular atlas A. This formal
drawback is eliminated with the help of the following lemma.
Lemma 5.1. Suppose A ⊂ M is an open set in M as defined above. If ψ : V → U is a chart for M , in
general not belonging to A, but compatible with all the charts in A, then the set ψ −1 (A ∩ U ) ⊂ V ⊂ Rn
is open.
Proof. Consider the sets A ∩ U ⊂ M and ψ −1 (A ∩ U ) ⊂ V ⊂ Rn . We can represent the latter set as
follows: since the sets Uα cover M , we have
[
A=
A ∩ Uα ,
α
so
A∩U =
[
A ∩ Uα ∩ U ,
α
1
Students not familiar with basic topological notions should consult standard textbooks such as General
Topology by Kelley, Topology by Dugundji, or, among more modern (and elementary) texts, Basic Topology by
Armstrong. See also Appendix to this section. Possibly useful may be my notes for the lecture course Introduction to Topology that I taught some years ago; see http://www.maths.manchester.ac.uk/~tv/Teaching/
Topology353/Fall2005/index.html
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Theodore Voronov.
and
!
ψ −1 (A ∩ U ) = ψ −1
[
A ∩ Uα ∩ U
α
=
[
ψ −1 (A ∩ Uα ∩ U ) .
α
Each ψ −1 (A ∩ Uα ∩ U ) is a subset of ψ −1 (Uα ∩ U ) ⊂ V ⊂ Rn . If we show that it is an open set in V
(or Rn ) for each α, then the union will also be open and so will ψ −1 (A ∩ U ), which is as claimed. Now,
n
−1
n
we may consider the sets ϕ−1
α (Uα ∩ U ) ⊂ Vα ⊂ R and ϕα (A ∩ Uα ) ⊂ Vα ⊂ R . The former is open
by the definition of the compatibility of charts and the latter is open because A ⊂ M is an open set
according to the definition above (relative to the atlas A). Hence their intersection is open in Vα . By
the definition of the compatibility of charts, we also have that
−1
ψ −1 ◦ ϕα : ϕ−1
α (Uα ∩ U ) → ψ (Uα ∩ U )
is a diffeomorphism of open domains of Rn . In particular, ψ −1 ◦ ϕα maps open sets onto open sets. We
have
−1
(U
∩
U
)
∩
ϕ
(A
∩
U
)
= (ψ −1 ◦ ϕα ) ϕ−1
(ψ −1 ◦ ϕα ) ϕ−1
α
α
α
α
α (A ∩ Uα ∩ U ) =
ψ −1 (A ∩ Uα ∩ U ) ,
which is, therefore, an open subset of V ⊂ Rn as the image of an open subset of Vα ⊂ Rn under a
diffeomorphism (in particular, a homeomorphism). Hence the union of all ψ −1 (A ∩ Uα ∩ U ) over α is
open, and it is ψ −1 (A ∩ U ).
Therefore the notion of an ‘open set’ defined by an atlas specifying a manifold structure is the
same for all equivalent atlases, so the topology of a manifold M is determined by the manifold
structure itself and does not depend on a choice of a particular atlas that may be used.
Suppose there is already some topology on the set M . Under which condition will it coincide
with the manifold topology?
Theorem 5.2. A topology τ on a manifold M coincides with the manifold topology if and only if
for some atlas A = (ϕα , Vα , Uα ) all sets Uα are open in τ and all maps ϕα are homeomorphisms
w.r.t. τ and the usual Euclidean topology on Vα .
Proof. Consider the manifold topology. For each α, we need to see that the set Uα is open in this
topology. (See Example 5.1.) For that, we need to show that for all β, the sets ϕ−1
β (Uα ∩ Uβ ) ⊂ Vβ
are open. This is a part of the definition of a manifold (see Definition 1.1 in §1). We also need to see
that, for each α, the map ϕα : Vα → Uα is a homeomorphism (w.r.t. to the manifold topology on M
and the usual topology of Vα ⊂ Rn ). We need to show that both ϕα and ϕ−1
α are continuous. For
ϕα : Vα → Uα , consider a subset U ⊂ Uα open in the manifold topology. By the definition, that implies
−1
that ϕ−1
ϕα is continuous. For ϕ−1
α (U ∩ Uα ) = ϕα (U ) is open in Vα . Therefore
α : Uα → Vα , consider
−1
−1
an open subset V ⊂ Vα . We need to check that ϕα
V = ϕα (V ) is open in the manifold topology,
which is, by the definition, the condition that for all β, the set ϕ−1
β (ϕα (V ) ∩ Uβ ) ⊂ Vβ is open in Vβ .
−1
−1
But ϕβ (ϕα (V ) ∩ Uβ ) = (ϕβ ◦ ϕα ) V and we can use the condition that ϕ−1
β ◦ ϕα is a homeomorphism
of open domains of Euclidean spaces, which follows from the definition of a manifold (Definition 1.1 in
§1). Homeomorphisms map open sets to open sets, hence ϕ−1
β (ϕα (V ) ∩ Uβ ) is open. Therefore both ϕα
−1
and ϕα are continuous, so are mutually inverse homeomorphisms as claimed.
Conversely, let τ be some topology on M such that for all α the sets Uα are open in τ and the
maps ϕα are homeomorphisms w.r.t. τ . We need to show that τ coincides with the manifold topology
on M . Let A ⊂ M be an open set in τ . Then for all α, the set A ∩ Uα is open, and since ϕα is
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a homeomorphism, the set ϕ−1
α (A ∩ Uα ) is also open. That means that A is open in the manifold
topology. Now let a set A ⊂ M be open in the manifold topology. That means that ϕ−1
α (A ∩ Uα ) is
n
−1
open in Vα ⊂ R for all α. Since the maps ϕa (and thus ϕα ) are homeomorphisms with respect to τ ,
it follows that all the intersections A ∩ Uα are open in τ . By writing
[
A=
A ∩ Uα ,
α
we obtain that the set A is open in τ . Hence the two topologies coincide.
Example 5.2. The manifold topology of S n ⊂ Rn+1 coincides with the subspace topology.
Indeed, consider the subspace topology on S n (in which the open sets are the intersections with
S n of the open sets in Rn+1 ). Consider the atlas for S n consisting of the two charts given by the
stereographic projections from the north pole and the south pole. Since both sets S n \ {N } and
S n \ {S} are open in this topology, and the formulas for stereographic projection are given by
rational functions (for both direct and inverse maps), then by Theorem 5.2, hence continuous,
we may conclude that the subspace topology and the topology induced by the manifold structure
for S n coincide.
What is the relation between smoothness and continuity for maps between manifolds? Recall
that for a map of topological spaces, the continuity is equivalent to the continuity at each point
defined in terms of neighborhoods. So it is a local property. This is used in the following theorem.
Theorem 5.3. Every smooth map F : M1 → M2 between manifolds is continuous.
Proof. We know that smooth maps between (open domains of) Euclidean spaces are automatically
continuous. This is established in the course of differential calculus and basically follows from the
definition of differentiability and the ε-δ-definition of continuity. Fix atlases for M1 and M2 . Recall
that a map of manifolds F : M1n → M2m is called smooth if for all α and µ the sets (F ◦ ϕ1α )−1 (U2µ ) ⊂
n
V1α ⊂ Rn are open and the maps ϕ−1
2µ ◦ F ◦ ϕ1α between open domains of Euclidean spaces R and
Rm are smooth. Here ϕ1α : V1α → U1α are the charts for M1 and ϕ2µ : V2µ → U2µ are the charts
for M2 . In particular the maps ϕ−1
2µ ◦ F ◦ ϕ1α are continuous. By Theorem 5.2, the maps ϕ1α and
ϕ2µ are homeomorphisms. In particular,
for each α and µ the map ϕ1α restricted to the open set
−1
−1
−1
(F ◦ ϕ1α ) (U2µ ) = ϕ1α F (U2µ ) maps it homeomorphically onto the subset F −1 (U2µ ) ∩ U1α ⊂ U1α ,
which is thus an open subset of U1α and of M1 . Therefore we have an open cover of M1 by the sets
F −1 (U2µ ) ∩ U1α , and
the restrictions
of F on all elements of this cover are continuous because they can
−1
be written as ϕ2µ ◦ ϕ−1
2µ ◦ F ◦ ϕ1α ◦ ϕ1α , so are the compositions of continuous maps. Hence the map
F : M1 → M2 is continuous.
5.2
Bump functions and partitions of unity
We have to admit an embarrassing fact: for a general manifold M , we do not have tools allowing
to show the existence of smooth functions defined everywhere on M (besides constants). On Rn
we have plenty of functions: first of all, the standard coordinate functions xi , then polynomials
and various other smooth functions of the variables x1 , . . . , xn . By contrast, in the absence of
global coordinates on a manifold M n , how one can find a non-trivial smooth function?
If, however, a manifold M n can be embedded into some RN as a “multidimensional surface”
(more precisely, as a closed submanifold), then there are plenty of smooth functions: all restrictions onto M n of the standard coordinates y 1 , . . . , y N of RN . Conversely, if there are enough
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smooth functions on a manifold M n to separate points, then it is at least intuitively clear that
there is an embedding of M n into a Euclidean space of a large dimension.
So the questions about having “enough smooth functions” and about the possibility to embed
a manifold into a RN are closely related.
Let us make the following observation. Every topological space that can be realized as a
subspace of RN must satisfy some a priori topological conditions. The space RN has certain
properties inherited by all its subspaces.
Recall that a topological space is Hausdorff if for any two points x 6= y it is possible to find
open sets Ux 3 x and Uy 3 y s.t. Ux ∩ Uy = ∅.
Proposition 5.1. Any subspace of RN is Hausdorff.
Proof. Indeed, RN itself is Hausdorff, and any subspace of a Hausdorff space is Hausdorff.
Recall that a base of a topological space is a family of open sets such that an arbitrary open
set is the union of sets from that family.
Example 5.3. Suppose X is a metric space (e.g., Rn ). The collection of all open balls is a base.
Example 5.4. For Rn it is possible to show that it is sufficient to take all open balls with
rational centers (the coordinates of the centers must be rational numbers) and rational radii.
This base is countable because Q is countable.
A topological space is called second-countable if it has a countable base2 .
Proposition 5.2. Any subspace of RN is second-countable.
Proof. Indeed, as we have seen, RN is second-countable, and any subspace of a second-countable
space is second-countable.
Proposition 5.3. A manifold M n is second-countable if and only if it has a countable atlas.
Proof. Indeed, suppose M n is second-countable. That means there is a countable base B = (Bk ).
Consider an arbitrary atlas (ϕα : Vα → Uα ) on M n . Then every set Uα , which is open, is the
union of some open sets Bk ∈ B; hence these particular Bk ’s admit local coordinate systems as
the restrictions of the coordinate system existing on the domain Uα . Thus we arrive at a new
atlas whose charts have certain elements of the family B = (Bk ) as the codomains. In particular,
it is countable. Conversely, suppose on a manifold M n there is a countable atlas. To construct a
countable base for M n , take countable bases for the domains of each chart (as an open subset of
Rn ) and consider their images in M n . It is a countable family of open sets in M n and every open
set in M is the union of some elements of this family (since it is the union of its intersections
with the chart codomains).
Example 5.5. A compact manifold is always second-countable. Indeed, a compact manifold
admits a finite atlas and we may argue as above.
In view of the discussion above about the existence of non-constant smooth functions and
the possibility of realizing a manifold as a subspace in a higher-dimensional Euclidean space, we
shall make the following amendment to our original definition of manifolds.
2
This traditional terminology comes from certain “first” and “second” countability conditions considered in
genera topology.
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Differentiable Manifolds. Autumn 2014
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Definition 5.2 (Amendment to the definition of a manifold). From now on, we require that
any manifold as a topological space should be Hausdorff and possess a countable
base (i.e., be second-countable).
Remark 5.1. These additional requirements are not vacuous, i.e., do not follow from the existence of local charts. The existence of coordinate charts, i.e, a locally Euclidean structure for
a topological space is its local topological property, while the Hausdorff condition or secondcountability are independent global topological restrictions. It is possible to construct examples
of topological spaces that are locally Euclidean (can be covered by open sets homeomorphic
to open domains in Rn ) and even with a smooth structure, but that are not Hausdorff or not
second-countable. The famous counterexamples are in dimension one: the so-called “line with a
double origin” (defined as the identification space of R ∪ R where all the corresponding points
of the two lines are identified except for the two origins; the result is locally homeomorphic
to R, second-countable, but non-Hausdorff) and the so-called “Alexandroff long line” (Hausdorff, connected, locally homeomorphic to R, but not second-countable; we skip details of its
construction). We shall not discuss them further and refer the reader to topology textbooks.
With these conditions imposed on manifolds it is possible to show that there is plenty of
non-trivial smooth functions. Moreover, it is possible to show now that every manifold can be
embedded into a Euclidean space of sufficiently large dimension.
The following constructions give important technical tools. [The proofs are required only for
the enhanced version of the course.]
Recall that for any topological space X, the support of a function
f: X →R
(notation: Supp f ) is the closure of the subset where the function does not vanish:
Supp f = {x ∈ X | f (x) 6= 0} .
A function f has compact support if Supp f is compact. We also say that f is compactly supported
in an open set U if Supp f is compact and Supp f ⊂ U .
Theorem 5.4. For every point x ∈ M of a smooth manifold M there is a non-negative smooth
function gx ∈ C ∞ (M ) compactly supported in a neighborhood of x and which is identically 1 on
a smaller neighborhood.
Such functions are called bump functions. (If we need to emphasize the role of the point
x, we speak about “bump functions centered at x”.) A proof of their existence is based on a
construction for Rn and the Hausdorff condition.
Lemma 5.2 (Bump functions for R). There is a non-negative C ∞ -function on R that equals 1
on the segment [−1, 1] and equals 0 outside of the segment [−2, 2].
Proof. Consider the function f (x) defined as
(
e−1/x
f (x) :=
0
5
for x > 0
for x 6 0 .
Differentiable Manifolds. Autumn 2014
Theodore Voronov.
It is C ∞ (check!). Note that f (x) + f (1 − x) > 0 for all x. Define
h(x) :=
f (x)
.
f (x) + f (1 − x)
Since f (1 − x) = 0 for x > 1, the non-negative function h is identically 0 for x 6 0 and identically 1 for
x > 1. Then the function
g(x) := h(x + 2)h(2 − x)
is zero for x > 2 or x 6 −2 and equals 1 for |x| 6 1.
Lemma 5.3 (Bump functions for Rn ). Let Ca be a closed cube in Rn ,
Ca = (x1 , . . . , xn ) ∀i |xi | 6 a
(the closed coordinate cube with side 2a centered at the origin). There is a non-negative C ∞ function that equals 1 on C1 and equals 0 outside of C2 .
Proof. Define a function g : Rn → R as the product
g(x1 , . . . , xn ) := g(x1 )g(x2 ) . . . g(xn ) ,
where the function of a single variable g(xi ) at the RHS, for each i = 1, . . . , n , is the function constructed
in Lemma 5.2. The function g(x1 , . . . , xn ) has the required properties.
There is nothing special, of course, in the numbers 1 and 2: a function with the properties
such as in Lemma 5.3 can be constructed for arbitrary pairs of closed cubes Ca ⊂ Cb where
b > a > 0. In particular, the cubes can be taken as small as necessary.
Remark 5.2. The possibility to construct functions that interpolate as above between different
constant values taken at whole intervals is a crucial feature of the C ∞ setup. It is impossible to
achieve that with analytic functions. Analytic functions are “rigid”, while smooth functions are
“soft”.
The existence of bump functions for an arbitrary manifold follows from a construction for
Rn combined with the Hausdorff property.
Proof of Theorem 5.4. Consider a manifold M n and a point x ∈ M n . Consider a coordinate neighborhood U around x. Lemma 5.3 implies the local existence of a function with the required properties:
the existence of a function gx ∈ C ∞ (U ) (i.e., defined on the neighborhood U ) such that it is identically
1 near x and its support is contained in some open set O ⊂ U homeomorphic to an open cube in Rn .
In particular, the closure Ō of O is homeomorphic to a closed cube, hence is compact. We now extend
the function gx from the open set U to the whole M , by zero:
gx (y) := 0
for all y ∈ M \ U .
Now we have a function gx : M → R. All the required properties are satisfied, but we need to check
that gx ∈ C ∞ (M ), i.e., that the extended function remains smooth. (We are given the smoothness
at each point of U , but we need to check the smoothness at the points of M \ U .) It is sufficient to
show that for any y ∈ M \ U there is a whole open neighborhood W 3 y on which gx is identically
zero. Fix some y. Consider an arbitrary point z ∈ Ō. Since M is Hausdorff, there are disjoint open
neighborhoods Oyz 3 y and Ozy 3 z of y and z respectively. The collection {Ozy } indexed by z ∈ Ō
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Theodore Voronov.
(the point y is fixed) is an open cover of Ō. Since Ō is compact, we can extract
a finite number of open
T
sets {Ozk y }, k = 1, . . . , N , covering Ō. Then the finite intersection W = Oyzk of the corresponding
neighborhoods of y is an open neighborhood of y and does not intersect with any of the open sets
{Ozk y }, hence with their union and with the set Ō contained in that union. In fact, it is sufficient to
know that W 3 y does not contain any point where gx is non-zero. Hence the function gx is identically
zero on W . Therefore gx is smooth at y, and this is for each y in M \U . So γx ∈ C ∞ (M ) as desired.
Remark 5.3. From the proof it follows that it is possible to construct a bump function gx
compactly supported in any given neighborhood of x ∈ M n .
The existence of bump functions on a manifold implies that there are many globally defined
smooth functions. Moreover, an arbitrary smooth function defined on an open set can be extended (without changing it on a smaller open set) to a smooth function defined on the whole
manifold. Indeed, we can multiply it by a suitable bump function compactly supported inside a
smaller open set and then extend it by zero. This is used very often.
There is another tool related to bump functions, of even greater importance. It is the so called
‘partitions of unity’. In accordance with its direct meaning, this term stands for a decomposition
P
of the function identically equal to 1 into the sum of some (non-constant) functions as 1 = µ fµ .
It is crucial that the summands in a partition can be chosen compactly supported in open sets.
This allows to “localize” various global objects on a manifold by multiplying by a suitable
partition of unity and thus reducing them to the sums of objects each “living” inside its own
open set.
To introduce partitions of unity, we need some more definitions.
A collection of subsets Aµ ⊂ X of a topological space X is said to be locally finite if each
point x ∈ X has a neighborhood with the empty intersection with all Aµ except for a finite
number of indices µ.
Definition 5.3. A continuous partition of unity on a topological space X is a collection of
non-negative continuous functions (fµ ) with an arbitrary set of indices such that the collection
of their supports (Supp fµ ) is locally finite and for all x ∈ X,
X
fµ (x) = 1.
(1)
µ
(For each point the sum is actually finite.)
A partition of unity (fµ ) is called subordinate to a cover (Uα ) if for each µ there is an open
set Uα , where α = α(µ), such that Supp fµ ⊂ Uα . Here the collection of functions (fµ ) and the
collection of sets (Uα ) are each indexed by its own set of indices and a map µ 7→ α = α(µ) is
given.
We say that a partition of unity (fα ) is subordinate to an open cover (Uα ) with the same set
of indices if these index sets coincide and Supp fα ⊂ Uα for each α.
For smooth manifolds, we shall look for smooth partitions of unity, i.e., consisting of smooth
functions.
Theorem 5.5. For every open cover (Uα ) of a smooth manifold M , there exists a smooth partition of unity subordinate to this cover. Moreover, this partition of unity can be chosen with an
additional property: either
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1. the functions in the partition have compact support and the partition is countable, (fk )∞
k=1 ;
or
2. the partition (fα ) is subordinate to (Uα ) with the same set if indices and only the countable
number of functions fα are not identically zero.
A proof of Theorem 5.5 uses bump functions (in particular, the Hausdorff property) and the condition of second-countability.
Note that for a compact manifold M , all functions on M have compact support, so the difference
between the two cases in the theorem practically disappears. Also, since there is a finite sub-cover of
every open cover, the theorem is equivalent to the existence, for an arbitrary open cover of M , of a
partition of unity subordinate to it with the same set of indices, automatically consisting of functions
with compact support and such that only a finite number of functions are not identically zero. We shall
give a proof only for this case, i.e., when the manifold M is compact.
Let us emphasize that Theorem 5.5 holds for an arbitrary manifold M (of course, Hausdorff and
second-countable). Compactness is not necessary. However, we shall confine ourselves to compact
manifolds for two reasons. First, this is the case that we need most for the applications. Second, the
assumption of compactness of a manifold (superseding second-countability) simplifies the proof greatly.
Proof of Theorem 5.5 (for compact manifolds). Take an open cover (Uα ) of M . For each point xα ∈ Uα
consider a bump function gxα such that gxα = 1 on an open set Wαxα ⊂ Uα containing xα and
Supp gxα ⊂ Uα . Consider the collection (Wαxα ) indexed by α and xα as a new open cover of M . We
can extract from it a finite subcover. Denote its elements as Wk , k = 1, . . . , N , so that Wk = Wαk xk ,
where xk ∈ Uαk . Note that, for each k, the bump function gk := gαk xk equals 1 (and hence is positive)
on Wk . We define
gk
.
fk :=
g1 + . . . + gN
It is well-defined because all functions gP
k are non-negative and at each point of the manifold at least
one gk is positive. Clearly, fk > 0 and
gk = 1. Hence we have obtained a finite partition of unity.
Also, Supp fk = Supp gk ⊂ Uαk . To obtain a partition of unity with the same set of indices, consider
functions hα defined as fk for α = αk and as 0 otherwise.
Example 5.6. A partition of unity for the sphere S 2 . Use spherical coordinates. One can define
non-negative functions g1 (θ) and g2 (θ) such that g1 (θ) = 1 for 0 6 θ 6 π/3 and g1 (θ) = 0 for
2π/3 6 θ 6 π and, similarly, g2 (θ) such that g2 (θ) = 1 for 2π/3 6 θ 6 π and g2 (θ) = 0 for
0 6 θ 6 π/3. They are bump functions centered at the points N and S respectively. Note that
for each point of the sphere either g1 or g2 is positive3 . We set
f1 :=
g1
g1 + g2
and f2 :=
g2
.
g1 + g2
Hence (f1 , f2 ) is a partition of unity subordinate to the open cover (U1 , U2 ) where U1 = S 2 \ {S}
and U2 = S 2 \ {N }.
We shall use partitions of unity on manifolds as a tool for various constructions.
Among immediate topological consequences of the existence of partitions of unity is the following
statement.
3
Note that, unlike in the proof of Theorem 5.5, it is not true that one of the bump functions is identically 1
at each point, but we do not need that.
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Corollary 5.1 (Urysohn functions). Let C0 , C1 ⊂ M be closed subsets such that C0 ∩ C1 = ∅. There
is a function f ∈ C ∞ (M ) such that f ≡ 1 on C1 and f ≡ 0 on C0 .
Proof. Consider the open cover M = U0 ∪ U1 where Uα := M \ Cα and a subordinate partition of unity
f0 + f1 = 1. Take f := f0 . By definition, Supp f0 ⊂ U0 = M \ C0 . Thus f = 0 on C0 . Similarly, f1 = 0
on C1 . Thus f = f0 = 1 − f1 = 1 on C1 .
A function f constructed in Corollary 5.1 is called an Urysohn function for a pair of disjoint closed
sets C0 , C1 . On a topological space the existence of (continuous) Urysohn functions is a separation
property stronger than the Hausdorff condition.
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5.3
5.3.1
Embedding manifolds into RN
Immersions, submersions and embeddings
The maps dF (x) : Tx M → TF (x) N for all points x ∈ M assemble to a map of tangent bundles, denoted
dF or T F or DF or F∗ :
dF : T M → T N .
Properties of the tangent map dF can be used for studying properties of a smooth map F .
Definition 5.4. A map F is called a submersion or we say that F is submersive if the linear map
dF (x) is an epimorphism (i.e., surjective) at each point x. It is called an immersion (or an immersive
map) if the linear map dF (x) is a monomorphism (i.e., injective) at each point.
One can show, by using the implicit function theorem, that every immersion is “ locally” an injective
map. However, it is possible for a smooth map to be injective but not immersive. (Example: the map
R → R, x 7→ x3 .) A smooth map F : M → N that is both injective and immersive is called embedding.
Theorem 5.6. Suppose M is compact. Then the image of an embedding F : M → N is a closed
submanifold in N .
We do not give a proof. Compactness is necessary as shown by the following example.
√
Example 5.7. Consider F : R → T 2 given by F (t) = (e2πit , e2πi
show that the image is not a closed submanifold.
5.3.2
2t ).
It is an embedding, but one can
Possibility of embedding
Recall the following definition (briefly mentioned in subsection 5.3.1).
Definition 5.5. A (smooth) map of manifolds F : M n → N k is an embedding if it is injective and
all the tangent maps dF (x) : Tx M → Tf (x) N are also injective (that is, are monomorphisms of vector
spaces).
Theorem 5.7. For any compact manifold M n there is an embedding
F : M → RN ,
for a sufficiently large N .
Proof. Recall the construction of bump functions. Starting from an arbitrary atlas for M , we can
always construct a finite atlas with coordinate domains Uk , k = 1, . . . , p, having the following property:
there are smaller open sets Wk ⊂ Uk and non-negative functions gk such that each gk = 1 identically on
Wk and gk vanish on M \Uk and the sets Wk still give a cover of M . Let xak ∈ C ∞ (Uk ) be the coordinate
functions on Uk . Define the functions yka ∈ C ∞ (M ) as gk · xak on Uk and as zero on M \ Uk . (This is
well-defined, compare the construction of bump functions). Define a smooth map F : M → Rp(n+1) by
the formula
F : x 7→ (yka (x), gk (x)).
(2)
This map is injective. Indeed, suppose F (x) = F (y) for some x, y ∈ M . Since the sets (Wk ) cover M ,
there is some k such that gk (x) = 1. Hence gk (y) = 1. In particular, it follows that x and y are in the
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Theodore Voronov.
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same coordinate domain Uk (because gk = 0 on M \ Uk ). By definition of yka , then all their coordinates
there coincide, and hence x = y. Consider now the Jacobi matrix
a
∂yk ∂gk
,
,
∂xbl ∂xbl
which has n rows and (n + 1)p columns. For each x ∈ M , there is an open set Wk containing x where
∂xa
∂y a
gk = 1 identically. Thus ∂xkb = ∂xbk = δba (no summation over k!). Hence the rank of the Jacobi matrix
k
k
is n.
Thus any compact manifold can be considered as a submanifold of a Euclidean space RN for
sufficiently large N . This is true for non-compact manifolds as well, but the proof for arbitrary (not
necessarily compact) manifolds is more sophisticated.
5.3.3
Whitney Theorem
What is the dimension N ? Is it possible to minimize it?
We use the following statement4 without proof:
Lemma 5.4 (Corollary from Sard’s Lemma). A smooth map F : M n → N m cannot be surjective
if n < m.
Theorem 5.8 (Whitney). A manifold M n of dimension n can be embedded into RN with N = 2n + 1.
Proof. Suppose that an embedding M → RN is given. Is it possible to decrease the number N ? We
can identify M with its image in RN . Consider the orthogonal projection on the hyperplane l⊥ ⊂ RN
for some direction l ∈ RP N −1 . The projection is injective on M ⊂ RN if for any two distinct points
x and y in M , the vector x − y ∈ RN is not parallel to l. (That is, it does not belong to l.) In the
same way, the projection is injective on tangent vectors if no non-zero vector tangent to M is parallel
to l. Denote M̃ := M × M \ ∆ where ∆ = {(x, x)} is the ‘diagonal’. Denote T 0 M := T M \ M , i.e.,
the manifold of all non-zero tangent vectors. Consider the disjoint union M̃ ∪ T 0 M . We have a map
M̃ ∪ T 0 M → RP N −1 sending a pair (x, y) to the line spanned by x − y ∈ RN and a tangent vector
u 6= 0 to the line [u]. It is smooth and we can apply the corollary of Sard’s lemma. This map cannot be
surjective as long as 2n = dim(M̃ ∪ T 0 M ) < dim RP N −1 = N − 1, i.e., when N > 2n + 1. So it is always
possible (as long as N > 2n + 1) to find a direction of projection l onto the hyperplane l⊥ such that
its restriction on M is an embedding into l⊥ . Therefore, by considering projections onto hyperplanes
and applying the above argument many times, we see that the dimension of the ambient space can be
reduced to 2n + 1.
4
It belongs essentially to analysis, as the Sard Lemma itself. You can read about the Sard Lemma in the book
Modern Geometry, part 2, by Dubrovin, Novikov and Fomenko.
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Appendix. Topological notions
In this Appendix we recall basic topological notions that we may need. For details, please consult
textbooks such as, e.g., Armstrong, Dugundji or Kelley 5 (or any other suitable text treating basic
notions of set-theoretic topology). Obviously, the knowledge of the proofs is not required. Proofs,
where given, are provided only for you to see the logical connection between the concepts.
A topological space is a set endowed with a topology, while a topology on a set is defined as a family
of subsets such that it contains the empty set and the whole set as elements, and is closed under taking
arbitrary unions and finite intersections. Elements of a topology are referred to as open sets (w.r.t.
this topology). Topologies are often denoted by Greek lowercase letters such as τ . Closed sets are, by
definition, the complements of open sets. (They satisfy the ‘dual’ system of axioms: “arbitrary unions”
replaced by “arbitrary intersections” and “finite intersections” replaced by “finite unions”.) The closure
of any subset of a topological space is defined as the intersection of all closed sets that include the given
subset. (Dually, the interior of a subset is the union of all open sets that are included in the given
subset.)
There is a standard way of considering an arbitrary set as a topological space, by introducing the
so-called discrete topology, which, by definition, consists of all subsets. A set regarded as a topological
space in this way is called a discrete space.
Of course, the main point is that some sets possess topologies more interesting than discrete.
A topology may come from a metric structure on a set. This is a rather special case of topological
spaces, but appears frequently in applications. A metric or distance on a set is a function of pairs of
points of the set taking values in non-negative real numbers such that it is symmetric, vanishes for coinciding arguments only and satisfies the so-called ‘triangle inequality’ d(x, z) 6 d(x, y) + d(y, z). (Letters
such as d or ρ are typically used for denoting distance.) A set with a distance
function is called a metric
p
space. An example of a metric space is the set Rn where d(x, y) := (x1 − y 1 )2 + . . . + (xn − y n )2 .
(In a more invariant way, a metric such as above arises from a given norm on a vector space, as
d(x, y) := ||x − y||. We skip the general notion of a normed vector space. If there is an inner, or
scalar, product on a vector space, denoted as (v, u), then it defines the associated norm by the formula
||v||2 = (v, v). For example, the standard scalar product on Rn is defined in standard coordinates as
(v, u) = v 1 u1 + . . . + v n un , and a similar standard scalar product on Cn is defined by a similar formula
involving the complex conjugation, (v, u) = v̄ 1 u1 + . . . + v̄ n un . They in particular make Rn and Cn
normed vector spaces and hence, metric spaces.) If a metric space is given, open sets for it are defined
as follows: a set is called open if with every point it contains an ε-ball centered at this point; balls are
defined in the usual way, as the sets of all point of distance less than given (which is called the radius)
from a given point (called the center). One shows that balls themselves are open in the above sense,
and that so defined opens sets satisfy the axioms of a topology. Every open set for a metric space is
the union of open balls. For Rn with the metric as above, ‘balls’ are open balls in the usual geometric
sense. (One should note that for a finite-dimensional vector space such as Rn , all ‘natural’ ways of
introducing topology give the same, so it does not depend on a choice of a scalar product, for example,
or on standard coordinates used to define ‘balls’.)
For metric spaces, the notions of closed sets and the closure of a set can be described in terms of
limits of sequences and limit points. We refer to textbooks for details. Sometimes such a description is
more intuitively appealing. (“A set is closed if it contains all the limits of sequences in it”.)
In general, for a topological space, a family of open sets such that all open sets are unions of elements
of this family is called a base of topology. (A family of subsets of a set should satisfy certain properties,
5
Basic Topology, by M. A. Armstrong; Topology, by J. Dugundji; General Topology, by J. L. Kelley. Armstrong
is more elementary, while Kelley and Dugundji are more advanced. Good sources are also textbooks on analysis
or functional analysis.
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Theodore Voronov.
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which can be made axioms, to be a base for some topology, which is uniquely defined in this case.)
Many constructions can be made in terms of bases. An example of a base of the standard topology of Rn
is given by all open balls with rational radii and centers with rational (standard) coordinates. Clearly,
such a base is countable. A topological space possessing a countable base is called second-countable
and its existence is known as the ‘second countability axiom’.
According to the general philosophy of category theory, after having defined objects, we should
define the corresponding morphisms. For topological spaces, this is the notion of a continuous map. A
continuous map between two topological spaces is, by definition, a map of the underlying sets such that
the preimage of every open set is open. (Recall that, for a map f : X → Y and B ⊂ Y , the preimage
f −1 (B) ⊂ X is defined as f −1 (B) := {x ∈ X | f (x) ∈ B}.) An exercise is to show that replacing open
sets by closed sets in the definition of continuity gives the same notion; this is sometimes convenient.
One checks that an identity map is continuous and the composition of continuous maps is continuous.
Hence, there is a category whose objects are topological spaces and morphisms are continuous maps 6 .
An isomorphism in this category, i.e., a continuous map f : X → Y such that there exists a continuous
inverse map f −1 : Y → X is called a homeomorphism. (There should be no confusion between the
notation for preimages, where the map is not assumed to be invertible, and the notation for the inverse
map; however, if a map is invertible, the preimage of any subset is the same as the image under the
inverse map.) From mathematical analysis, a different approach to the continuity of maps is known;
it is based on the language of neighborhoods (and is an abstract version of the “ε-δ” language). A
(open) neighborhood of a point in a topological space is an arbitrary open set containing this point. A
particular example is an ε-neighborhood for metric spaces, which is the same as an ε-ball (i.e., a ball
of radius ε) with center at this point; so for the real line R we come back to the intervals (x − ε, x + ε)
familiar from the course of mathematical analysis. Now, a map f : X → Y between topological spaces
is called continuous at a point x0 if for each neighborhood V of f (x0 ) there is a neighborhood U of x0
such that f (U ) ⊂ V . It is proved that a map is continuous (in the sense of the above definition) if it is
continuous at all points.
According to the same general philosophy, if we have a category, then there are notions of ‘subobjects’ and ‘factor-objects’. We can say what are they for topological spaces.
An arbitrary subset A ⊂ X of a topological space X can be made a topological space; an open set
in A is defined as the intersection of an open set in X with A. This is a topology on A and the set
A together with it is called a subspace of X. (You may compare this simple situation with smooth
manifolds, for which, firstly, not all subsets can be naturally made manifolds, and, secondly, there are
different types of submanifolds, such as open submanifolds and closed submanifolds; there are actually
others that we have not discussed in these lectures.)
Let X be a topological space and a set Y = X/ ∼ is the quotient set w.r.t. an arbitrary equivalence
relation. The points of Y are the equivalence classes. (Alternatively, this can be described in terms of
a partition of X into pairwise disjoint subsets, which become the points of the set Y .) Then there is
a canonical way of introducing a topology into Y . Namely, a subset B ⊂ Y is called open if p−1 (B) is
open in X; here p is the projection map sending every point of X to its equivalence class in Y . The set
Y endowed with this topology is called an identification space of X. (For comparison with manifolds:
only very special quotients of a manifold can be endowed with a manifold structure. We do not discuss
quotients of manifolds in these lectures.)
Another important construction leading to new topological spaces, is that of a product. For topological spaces X and Y , their product X × Y as sets can be endowed with a topology as follows: one
takes as a base of the so-called product topology on X × Y all sets of the form U × V ⊂ X × Y where
6
It is clear that if two sets are endowed with the discrete topology, then arbitrary maps between them will be
continuous. Therefore, the category of sets and arbitrary maps is included as a ‘subcategory’ in the category of
topological spaces and continuous maps.
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Differentiable Manifolds. Autumn 2014
Theodore Voronov.
U ⊂ X and V ⊂ Y are open subsets of respective spaces; i.e., the product topology consists of all
unions of sets of the form U × V . It is possible to define products of several topological spaces and
actually of arbitrary families. Products of topological spaces are associative: the spaces (X × Y ) × Y
and X × (Y × Z) are naturally homeomorphic (and homeomorphic to the triple product X × Y × Z).
Among properties of topological spaces that are the most important for us are the Hausdorff property, compactness and connectedness.
A topological space is Hausdorff (or is a Hausdorff space) if every two distinct points have disjoint open neighborhoods, i.e., with the empty intersection. All metric spaces (in particular, Rn ) are
Hausdorff because as disjoint neighborhoods one can take open balls of sufficiently small radii. Every
subspace of a Hausdorff space is Hausdorff, and the product of Hausdorff spaces is Hausdorff. (However,
an identification space of a Hausdorff space is not necessarily Hausdorff.) The Hausdorff property is one
of several so-called ‘separation properties’ considered in general topology; possibly the most important.
An example of another strong separation condition is the existence, for every two disjoint closed subsets, of a continuous function that is identically 1 on one subset and is identically 0 on another subset;
this is called functional separability or Urysohn property. (As we showed, this holds automatically for
manifolds following for them from the Hausdorff condition and the second-countability.)
A topological space is called compact, if one can extract a finite subcover from its any open cover.
(An open cover is a family of open sets such that their union gives the whole space. It is traditional to
use capital Gothic letters for denoting open covers, e.g., U or V, which are ‘Gothic U ’ and ‘Gothic V ’.)
Note that a subspace S of a topological space X is compact if one can extractSa finite subcover from
any open cover of S in X, i.e., a collection of open sets Uα ⊂ X such that S ⊂ Uα .
For discrete topological spaces, being compact is equivalent to being finite. In general, compact
topological spaces can be perceived as an extension and generalization of the notion of a finite set.
Properties valid for finite sets in the ‘discrete setup’ often have their counterparts for compact spaces
in the ‘topological setup’.
Theorem. Any closed subspace of a compact space is compact.
Proof. Consider an open cover of S in X. Add X \ S to this collection. We obtain an open cover of X.
Extract a finite subcover and for aesthetical purposes throw away X \ S, if it is still present. This is a
finite subcover for S.
Theorem. Let f : X → Y be a continuous map, X compact. Then f (X) ⊂ Y is compact.
Proof. Let (Uα ) be an open cover of f (X). ThenSf −1 (Uα ) is anSopen cover of X.S Extract a finite
subcover f −1 (Uαk ), k = 1, . . . , N . Then f (X) = f ( f −1 (Uαk )) = f (f −1 (Uαk )) = Uαk .
Corollary. An identification space of a compact space is compact.
Theorem. If X and Y are compact, then the product X × Y is also compact.
Proof. See, for example, Armstrong, Theorem 3.15.
(Actually, the statement holds for the products of arbitrary families of compact spaces. This is
more complicated and is known as the Tychonoff Theorem.)
There is a non-trivial interaction between compactness and Hausdorff property.
Theorem. A compact subspace of a Hausdorff space is a closed set.
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Theodore Voronov.
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Proof. Consider S ⊂ X where X is Hausdorff and S is compact. To show that S is closed, we need to
show that X \ S is open. It is sufficient to see that every point x ∈ X \ S has a neighborhood U such
that U ⊂ X \ S (then the whole X \ S is the union of such neighborhoods). Fix a point x ∈ X \ S.
To construct a desired neighborhood for it, consider points s ∈ S; by the Hausdorff property of X,
it is possible to separate each of them from x ∈ X \ S by a pair of disjoint neighborhoods V(s) 3 s
and U(s) 3 x. In particular we have an open cover of S in X by the neighborhoods V(s) . Since S is
compact, we may take a finite subcover of S by neighborhoods V(sk ) where k = 1, . . . , N . To them
T
correspond the neighborhoods U(sk ) of x ∈ X \ S (a finite number). Their intersection N
k=1 U(sk ) =: U
is a neighborhood of x ∈ X \ S that does not intersect any of the neighborhoods V(sk ) covering S; in
particular, U does not intersect S, i.e., is entirely in X \ S as desired. Therefore X \ S is open and S
is closed.
Theorem (Homeomorphism Theorem). If a continuous map from a compact space to a Hausdorff space
is injective, then it is a homeomorphism on its image.
Proof. We may replace the codomain by the image (which is also Hausdorff as a subspace of a Hausdorff
space) and consider bijections from the start. The theorem then states that a continuous bijection
from a compact space to a Hausdorff space is a homeomorphism, i.e., the inverse map is continuous
automatically. We shall check the continuity in the form using closed sets: that is, that the preimage
of a closed set is closed. The preimage w.r.t. the inverse map is the same as the image w.r.t. the
initial map. Thus, we need to show that every closed set is mapped (by the given map) to a closed set.
Since the domain is compact, a closed subset of it is compact; its image under a continuous map is also
compact. Now, it is a compact subspace of the codomain, which is assumed to be Hausdorff. Hence,
by the previous theorem, it is a closed subset, which is exactly what we need. Therefore the inverse
map is continuous and we have a homeomorphism.
For metric spaces, it is possible to give an alternative approach to compactness characterizing
compact metric spaces in terms of the so-called ‘ε-nets’. (See, for example, in Liusternik and Sobolev,
Elements of Functional Analysis, § 9; or in many other advanced analysis textbooks.) We will not go into
that. It is clear that a compact metric space must be bounded (that is, all the distances between points
are bounded from above by a constant C). Indeed, consider concentric open balls (with an arbitrary
center); they make an open cover. By extracting a finite subcover, we see that the whole space is a
ball 7 of sufficiently large radius, so is bounded. One has the following criterion of compactness for
subspaces of RN .
Theorem (Heine–Borel Theorem). A subspace of RN is compact if and only if it is bounded (i.e.,
contained in a large ball or ‘box’) and closed.
Sketch of proof. In one direction the statement immediately follows from theorems already given: if
a subspace S ⊂ RN is compact, it has to be bounded (as any compact metric space) and closed (as
a compact subspace of a Hausdorff space). So we need to show the converse: if a subspace S ⊂ RN
is bounded and closed, it is compact. If S ⊂ RN is bounded, then S is contained in the closed cube
CA = {x | ∀i |xi | 6 A}, for A large enough. Since it is given that S is closed, it will be sufficient
to observe that any closed cube CA ⊂ RN is compact: then an arbitrary closed subset S ⊂ CA is
also compact. The cube CA is the product of closed intervals [−A, A] ⊂ R, and everything reduces
to showing that an arbitrary closed interval (closed segment) in R is compact. This is a classical
statement of the XIX century mathematical analysis due to H. Heine and E. Borel, which is frequently
referred to as the ‘Heine–Borel Lemma’. It reflects a crucial property of the real numbers, their so-called
7
Don’t be confused; this does not imply anything about the ‘shape’ of the space.
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Theodore Voronov.
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‘completeness’ (as opposed to the ‘incompleteness’ of the rational numbers). Various manifestations of
the completeness of R are used in different proofs of the Heine–Borel Lemma, such as the existence of
the supremum of a bounded set or the ‘nested intervals property’ (which can be applied directly to RN
as well). We do not discuss this further here, but refer instead to good calculus/ analysis textbooks or
the aforementioned book by Armstrong.
From here we see that Rn itself is non-compact, but, for example, S n is compact. Boundedness is
obvious. For closedness, we make a general remark that an arbitrary subset of RN specified by a system
of equations with the l.h.s. given by continuous functions is closed. (Indeed, it is the preimage of a
closed set — a single point — under a continuous map.) This applies to the sphere, so S n is compact by
the Heine–Borel theorem.
You should work out a good intuitive understanding which examples of manifolds (or more general
topological spaces) are compact and which not. For example, the sphere S n , the projective spaces
RP n and CP n , the orthogonal group O(n) and the unitary group U (n), are all compact spaces. Also
compact are a closed cube and a closed ball in Rn (the closure of an open ball), and the closed Möbius
band. At the same time, the sphere without a point (which is homeomorphic to Rn ), an open ball,
the open Möbius band, the non-negative half-space Rn+ = {x | xn 6 0}, the space of all straight lines
in Rn+1 (not necessarily through the origin; compare with the definition of RP n ), the general linear
group GL(n), the special linear group SL(n) and the Lorentz group arising in relativity theory, — are
all non-compact spaces.
Finally, we need to discuss connectedness or the question, ‘of how many pieces a topological space
consists’. In topology they distinguish two notions. One is called (just) ‘connectedness’ and the other,
‘path-connectedness’.
A topological space is called connected if it is impossible to present it as the union of two non-empty
disjoint open sets. (Otherwise it is disconnected.)
Theorem. The image of a connected space under a continuous map is connected.
Proof. Consider a continuous map f : X → Y . Suppose f (X) ⊂ Y is disconnected. That means that
f (X) = V0 ∪ V1 where both V0 and V1 are open, non-empty and V0 ∩ V1 = ∅. Consider the preimages
f −1 (V0 ) and f −1 (V1 ). They are non-empty (because V0 and V1 are subsets of f (X)) and open (because
f is continuous). Also, f −1 (V0 ) ∪ f −1 (V1 ) = X (indeed, each x ∈ X is mapped to some y either in V0
or V1 , so x must belong to one of the preimages). Suppose there exists x ∈ f −1 (V0 ) ∩ f −1 (V1 ). Then
f (x) ∈ V0 and f (x) ∈ V1 , which contradicts V0 ∩V1 = ∅. Hence f −1 (V0 )∩f −1 (V1 ) = ∅ and we conclude
that X is disconnected, which contradicts the assumption.
Corollary. An identification space of a connected space is connected.
(Obviously, a subspace of a connected space need not to be connected.)
Theorem. The product of connected spaces is connected.
Proof. See Armstrong, § 3.
A topological space is called path-connected (or linearly connected, or arcwise connected ) if every
two points of it can be joined by a path. (A path in X is a continuous map γ : [a, b] → X. To join
points x0 and x1 by a path means to find a path such that γ(a) = x0 and γ(b) = x1 . Without loss of
generality, the standard segment [0, 1] can be taken as the domain of definition of paths.)
One remarkable property of paths is that they can be ‘composed’: loosely, the end of one path is
taken as the beginning of the other path. The precise definition can slightly vary depending on whether
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Theodore Voronov.
Differentiable Manifolds. Autumn 2014
we wish to keep the unit segment [0, 1] as the standard domain or allow more general segments. In the
version where all paths are maps [0, 1] → X, we can set
(
γ1 (2t)
for t ∈ [0, 21 ] ,
(γ1 ∗ γ2 )(t) =
γ2 (2t − 1)
for t ∈ [ 21 , 1] ,
for paths γ1 : [0, 1] → X, γ2 : [0, 1] → X, where γ1 (1) = γ2 (0). Clearly, γ1 ∗ γ2 is well-defined a
continuous map [0, 1] → X. It is called the composition of γ1 and γ2 (which should not be confused
with the composition of maps). This operation is not associative (check!), but not very far from that.
There is also the notion of the inverse path, which is defined as γ −1 (t) = γ(1 − t). In particular, the
composition of paths and the inverse paths can be used for the following remark: for a space X to be
path-connected, it is necessary and sufficient that all points can be joined with one particular point
x0 ∈ X arbitrarily chosen as an ‘origin’.
There are analogs for path-connected spaces of the statements concerning connected spaces given
above. Their proofs are simpler and we leave them as exercises.
Theorem. The image of a path-connected space under a continuous map is connected.
Proof. Exercise for the reader.
Corollary. An identification space of a path-connected space is path-connected.
Proof. Exercise for the reader.
Theorem. The product of path-connected spaces is path-connected.
Proof. Exercise for the reader.
The relation between path-connectedness and connectedness in the sense defined above is based on
the following key statement.
Lemma. The real line R and all open, closed or half-open intervals in it such as [a, b], (a, b], etc. (and
also infinite intervals) are connected.
Proof. See in Armstrong, § 3. Similarly to the classical Heine–Borel lemma (on the compactness of
the closed interval), this is a manifestation of the completeness of the real numbers; namely, it can be
deduced from the existence of the precise upper bounds (suprema) for arbitrary bounded sets in R.
Corollary. A path-connected topological space is connected.
Proof. Suppose X is path-connected. If it is disconnected, we can present it as X = U0 ∪ U1 with open,
non-empty and disjoint U0 and U1 . Take xo ∈ U0 and x1 ∈ U1 and join them by a path γ : [0, 1] →
X. Consider the preimages γ −1 (U0 ) and γ −1 (U1 ). They are non-empty (because 0 ∈ γ −1 (U0 ) and
1 ∈ γ −1 (U1 )), open (because γ is continuous), their union is the whole segment [0, 1], and they are
disjoint. (Check! Everything is very similar to the proof of the theorem about the image of a connected
space.) We arrive to the absurd statement that [0, 1] is disconnected. Therefore the space X has to be
connected.
So path-connectedness implies connectedness. The converse is wrong as shown by the standard
counterexample: the topological space defined as a subspace of the plane and obtained as the union of
a part of the graph y = sin x1 for x ∈ (0, π2 ] and the vertical segment [−1, 1] of the y-axis. It is possible
to see that the result is connected but not path-connected (one cannot join a point on the graph with
a point on the vertical segment by a path). See details in topology textbooks.
However, if a topological space is well-behaved locally (unlike the above example), connectedness
for it implies path-connectedness. More precisely, one needs that a space be locally path-connected,
which means, by definition, that every point has a path-connected neighborhood.
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Theodore Voronov.
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Theorem. A connected and locally path-connected space is path-connected.
Proof. Suppose X is connected and locally path-connected. Fix a point a ∈ X and define H as the set
of all points that can be joined with a. The aim is to prove that H = X. Consider a point h ∈ H.
There is a neighborhood Uh of h in X which is path-connected. In particular, every point of it can
be joined with h, hence with a. Hence Uh ⊂ H for every h ∈ H. It follows that H is open. Consider
X \ H. Suppose it is non-empty. Take k ∈ X \ H. There is an open neighborhood Uk of k in X which
is path-connected. In particular, every point x of it can be joined with k. If x could also be joined with
a, then k could be joined with a, which is a contradiction. Hence no point of Uk can be joined with a,
i.e. Uk ⊂ X \ H. That means that X \ H is open. This is a contradiction with the connectedness of X!
It follows that X \ H has to be empty, i.e., X = H.
The space Rn and its convex 8 subspaces such as open balls are path-connected (use straight line
segments as paths). Therefore all manifolds (and arbitrary locally Euclidean spaces, without the assumptions of second-countability or being Hausdorff) are locally path-connected, and for them the
notions of connectedness and path-connectedness coincide. For manifolds, we are free not to make a
distinction; we call everything ‘connectedness’, but because path-connectedness is simpler speak mainly
about the latter.
Examples of connected manifolds include the sphere S n (for n > 0), the projective spaces RP n and
n
CP for all n (when n = 0 they are just points and for n > 0 they are continuous images of spheres),
the unitary group U (n), the special orthogonal group SO(n), the special linear group SL(n), and the
complex general linear group GL(n, C) (but not the real version!).
Examples of disconnected manifolds include the 0-sphere S 0 = {−1, +1}, the orthogonal group
O(n), the general group GL(n), the Lorentz group O(1, 3) and, more generally, the pseudo-orthogonal
group O(p, q). Except for the Lorentz group and the pseudo-orthogonal group O(p, q), the “number of
connected pieces” in each of the examples equals 2 (for example, the matrices with positive determinant
and the matrices with negative determinant). The Lorentz group O(1, 3) has 4 “connected pieces” and
so does the group O(p, q) if pq > 0.
8
Recall that a subset of Rn is called convex if with any two points it contains the whole segment joining these
points.
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