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General Topology
1
Metric spaces
Definition 1.1. A metric space is a set together with a function
dist : X × X → R,
called a metric, such that the following three laws are satisfied:
1. (positivity) dist(x, y) ≥ 0 with equality if and only if x = y,
2. (symmetry) dist(x, y) = dist(y, x), and
3. (triangle inequality) dist(x, z) ≤ dist(x, y) + dist(y, z).
In a metric space we define the ε-ball, ε > 0, about a point x ∈ X to be
Bε (x) = { y ∈ X | dist(x, y) < ε }.
Also, a subset U ⊆ X is said to be open if, for each point x ∈ U , there is
an ε-ball Bε (x) about x such that Bε (x) ⊆ U . A subset is called closed if its
complement is open.
Definition. A function f : X → Y between metric spaces is continuous if for
every ε > 0 there exists δ > 0 such that, for every x, y ∈ X,
distX (x, y) < δ =⇒ distY (f (x), f (y)) < ε.
Proposition 1.2. A function f : X → Y between metric spaces is continuous
if and only if f −1 (U ) is open in X for each open subset U of X.
Proposition 1.3. If dist1 and dist2 are metrics on the same set X which satisfy
the hypothesis that for any point x ∈ X and ε > 0 there is a δ > 0 such that
dist1 (x, y) < δ =⇒ dist2 (x, y) < ε
and
dist2 (x, y) < δ =⇒ dist1 (x, y) < ε,
then these metrics define the same open sets in X.
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Topological spaces
Definition 2.1. A topological space is a set X together with a collection of
subsets of X called open sets such that:
1. the intersection of two open sets is open,
2. the union of any collection of open sets is open, and
3. the empty set ∅ and the whole space X are open.
Additionally, a subset C ⊆ X is called closed if its complement X − C is
open.
Remark. The collection of open sets itself is usually called topology (on X).
Definition 2.2. If X and Y are topological spaces and f : X → Y is a function,
then f is said to be continuous if f −1 (U ) is open in X for each open set U in
Y.
Remark. In the topological setting, the term map is used exclusively for continuous function. In other settings the term map may have different meaning.
Definition 2.3. If X is a topological space and x ∈ X, then a set N is called
a neighborhood of x if there is an open set U such that x ∈ U ⊆ N .
Definition 2.4. If X is a topological space and x ∈ X, then a collection Bx of
subsets of X is called a neighborhood basis at x if
1. each member of Bx is a neighborhood of x and
2. each neighborhood of x contains some member of Bx .
Definition 2.5. A function f : X → Y between topological spaces is continuous
at x, where x ∈ X, if, given any neighborhood N of f (x) in Y , there is a
neighborhood M of x in X such that f (M ) ⊆ N .
Proposition 2.6. A function f : X → Y between topological spaces is continuous if and only if it is continuous at each point x ∈ X.
Definition 2.7. A function f : X → Y between topological spaces is a homeomorphism if f −1 : Y → X exists (i.e., f is bijection) and both f and f −1 are
continuous.
Remark. The notation X ≈ Y means that X is homeomorphic to Y , i.e., there
exists a homeomorphism f : X → Y .
Definition 2.8. If X is a topological space and B is a colection of subsets of X,
then B is called a basis for the topology of X if the open sets are precisely the
unions of members of B (in particular the members of B are open). A collection
of S of subsets of X is a subbasis for the topology of X if the set B of finite
intersections of members of S is a basis.
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Definition 2.9. A topological space is said to be first countable if each point
has a countable neighborhood basis.
Definition 2.10. A topological space is said to be second countable if its topology has a countable basis.
Definition 2.11. A sequence f1 , f2 , f3 , . . . of continuous functions from a topological space X to a metric space Y converges uniformly to a function f : X → Y
if, for each ε > 0, there exists a number n such that
dist(fi (x), f (x)) < ε,
for all i > n and all x ∈ X.
Proposition 2.12. If a sequence f1 , f2 , f3 , . . . of continuous functions from
a topological space X to a metric space Y converges uniformly to a function
f : X → Y , then f is continuous.
Proposition 2.13. A function f : X → Y between topological spaces is open
if f (U ) is open in Y for each open subset U in X. It is closed if f (C) is closed
in Y for each closed subset C in X.
Definition 2.14. Let T be a topology on X that satisfies the condition P . The
topology T is the smallest (coarsest, weakest) topology on X with property P
if T ⊆ T 0 , for any other topology T 0 on X that has the property P . It is the
largest (finest, strongest) topology on X with property P if T 0 ⊆ T , for any
other topology T 0 on X that has the property P .
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Subspaces
Definition 3.1. If X is a topological space and A ⊆ X, then the relative
topology or the subspace topology on A (with respect to X) is the collection of
intersections of A with the open subsets of X. With this topology A is called a
subspace of A.
Proposition 3.2. If Y is a subspace of X then A ⊆ Y is closed in Y if and
only of A = Y ∩ Y for some closed subset C of X.
Proposition 3.3. If X is a topological space and A ⊆ X, then there exists a
largest open set U with U ⊆ A. This set is called the interior of A and denoted
by int(A).
Proposition 3.4. If X is a topological space and A ⊆ X, then there exists
a smallest closed set F with A ⊆ F . This set is called the closure of A and
denoted by A.
Proposition 3.5. If A ⊆ Y ⊆ X then A
Y
X
X, A = A .
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Y
=A
X
∩ Y . Thus if Y is closed in
Definition 3.6. If X is a topological space and A ⊆ X, then the boundary (or
frontier ) of A is ∂A = (A) ∈ X − A.
Proposition 3.7. If Y ⊆ X then the set of intersections of Y with members of
a basis of X is a basis of the relative topology of Y .
Proposition 3.8. If X, Y, Z are topological spaces, Z is a subspace of Y and
Y is a subspace of X, then Z is a subspace of X.
Proposition 3.9. If X is a metric space and A ⊆ X, then A coincides with
the set of limits in X of sequences of point in A.
Definition 3.10. A subset A of a topological space X is called dense if A = X.
It is called nowhere dense if intA = ∅.
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Connectivity and components
Definition 4.1. A topological space is connected if it is not the disjoint union
of two nonempty open subsets.
Definition 4.2. A subset A of a topological space X is clopen if it is both open
and closed. in X.
Proposition 4.3. A topological space X is connected if and only if its only
clopen subsets are and X.
Definition 4.4. A discrete valued map on a topological space X is any map
from X to a discrete space D.
Proposition 4.5. A topological space X is connected if and only if every discrete valued map on X is constant.
Proposition 4.6. If X → Y is continuous and X is connected, then f (X) is
connected.
Proposition 4.7. If {Yi }i∈I is a collection of connected sets in a topological
space X and no two of the Yi are disjoint, then ∪i∈I Yi is connected.
Corollary 4.8. The relation “p and q belong to a connected subset of X” is an
equivalence relation on X.
Definition 4.9. The equivalence classes of the equivalence relation in the last
corollary are called the connected components of X.
Definition 4.10. Components of space X are connected and closed. Each
connected set is contained in a connected component (thus the components are
maximal connected subsets). Components are either equal of disjoint and fill
out X.
Proposition 4.11. The statement “d(p) = d(q) for every discrete valued map
d on X” is an equivalence relation.
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Definition 4.12. The equivalence classes of the equivalence relation in the last
proposition are called the quasi-components of X.
Definition 4.13. Quasi-components of space X are closed. Each connected set
is contained in a quasi-component (in particular each component is contained
in a quasi-component). Quasi-components are either equal of disjoint and fill
out X.
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Separation Axioms
Definition 5.1. The separation axioms:
(T0 ) A topological space X is T0 if, for any two points x 6= y in X, there is an
open set containing one point but not the other.
(T1 ) A topological space X is T1 if, for any two points x 6= y in X, there is an
open set containing x but not y.
(T2 ) A topological space X is T2 (or Hausdorff ) if, for any two points x 6= y
in X, there are disjoint open sets U and V with x ∈ U and y ∈ V .
(T3 ) A T1 topological space X is T3 (or regular ) if, for any point x in X and
any closed set F not containing x, there are disjoint open sets U and V
with x ∈ U and F ⊆ V .
(T4 ) A T1 topological space X is T4 (or normal ) if, for any disjoint closed sets
F and G, there are disjoint open sets U and V with F ⊆ U and G ⊆ V .
Corollary 5.2. A Hausdorff space X is regular if and only if, for every point
x ∈ X, the closed neighborhoods of x form a neighborhood basis of x.
Corollary 5.3. A subspace of a regular space is regular.
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Nets (Moore-Smith convergence)
Definition 6.1. A directed set D is a partially ordered set such that, for any
two elements α and β of D, there is τ ∈ D with α ≤ τ and β ≤ τ .
Definition 6.2. A net in a topological set X is a directed set D together with
a function Φ : D → X.
Definition 6.3. If Φ : D → X is a net in the topological space X and A ⊆ X,
the Φ is frequently in A if, for every α ∈ D there exists β ∈ D such that α ≤ β
and Φ(β) ∈ A. The net Φ is eventually in A if there exists α ∈ D such that
Φ(β) ∈ A for all β ≥ α.
Definition 6.4. A net Φ : D → X in a topological space converges to x ∈ X
if, for every neighborhood U of x, Φ is eventually in U .
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Proposition 6.5. A topological space X is Hausdorff if and only if any two
limits of any convergent net are equal.
Proposition 6.6. A function f : X → Y between topological spaces is continuous if and only if, for every net Φ in X converging to x ∈ X, the net f ◦ Φ in
Y converges to f (x).
Proposition 6.7. If A ⊆ X, then A coincides with the set of limits of nets in
A which converge in X.
Definition 6.8. If D and D0 are directed sets and h : D0 → D is a function,
then h is final if, for every δ ∈ D, there exists δ 0 ∈ D0 such that
δ 0 ≤ α0 =⇒ δ ≤ h(α).
Definition 6.9. A subnet of a net µ : D → X is the composition µ ◦ h of µ
with a final function h : D0 toD.
Proposition 6.10. A net {xα } in X is frequently in each neighborhood of x ∈ X
if and only if it has a subnet which converges to x.
Definition 6.11. A net in a set X is universal if, for any A ⊆ X, the net is
either eventually in A or eventually in X − A.
Proposition 6.12. The composition of a universal net in X with a function
f : X → Y is a universal net in Y .
Theorem 6.13. Every net has a universal subnet.
Proposition 6.14. A subnet of a universal net is universal.
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Compactness
Definition 7.1. A covering of a topological space X is a collection of sets
whose union is X. It is an open covering if the sets in the collection are open.
A subcover is a subset of the collection that still covers the space.
Definition 7.2 (Heine-Borel property). A topological space X is compact if
every open covering of X has a finite subcover.
Definition 7.3. A collection of sets has the finite intersection property if the
intersection of every finite subcollection is nonempty.
Theorem 7.4. A topological space X is compact if and only if for every collection of closed subsets of X with the finite intersection property, the intersection
of the entire collection is nonempty.
Theorem 7.5. If X is a Hausdorff space, then every compact subset of X is
closed.
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Theorem 7.6. If f : X → Y is continuous and X is compact, then f (X) is
compact.
Theorem 7.7. If X is compact and A ⊆ X is closed, then A is compact.
Theorem 7.8. If X is compact, Y is Hausdorff, and f : X → Y is continuous
and bijective, then f is a homeomorphism.
Theorem 7.9. The unit interval I = [0, 1] is compact.
Theorem 7.10. A real valued map on a compact space assumes a maximum
value.
Theorem 7.11. A compact Hausdorff space is normal.
Definition 7.12. A map f : X → Y between topological spaces is proper is
f −1 (K) is compact for each compact subset K of Y .
Proposition 7.13. If f : X → Y is a closed map and f −1 (y) is compact for
each point y ∈ Y , then f is proper.
Theorem 7.14. For a topological; space X, the following are equivalent.
1. X is compact.
2. Every collection of closed subsets with the finite intersection property has
a nonempty intersection.
3. Every universal net in X is convergent.
4. Every net in X has a convergent subnet.
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Products
Proposition 8.1. The projections πX : X × Y → X and πY : X × Y → Y
are continuous and the product topology is the smallest for which this is true.
Similarly for the case of infinite products.
Proposition 8.2. If X is compact, the projection πY : X × Y → Y is closed.
Corollary 8.3. If X is compact, the projection πY : X × Y → Y is proper.
Corollary 8.4. If X and Y are compact, then X × Y is compact.
Corollary 8.5 (Tychonoff Theorem for Finite Products). If the Xi are compact,
then X1 × X2 × · · · × Xn is compact.
Corollary 8.6. The cube I n is compact.
Proposition 8.7. A net in a product space X = Πi∈I Xi converges to the point
(. . . , xi , . . . ) if and only if its composition with each projection πi : X → Xi
converges to xi .
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Theorem 8.8 (Tychonoff). The product of an arbitrary collection of compact
spaces is compact.
Proposition 8.9. A net {fα } in X A converges to f ∈ X A if and only if for
every x ∈ X, fα (x) → f (x). In particular, lima lpha(fα (x)) = (limα fα )(x).
Definition 8.10. If X and Y are topological spaces, then their topological sum
or disjouint union X + Y is the set X × {0} ∪ Y × {1} with the topology making
X × {0} and Y × {1} clopen and the inclusions x 7→ (x, 0) and y 7→ (y, 1) of X
and Y into X + Y , respectively, homeomorphisms to their images X × {0} and
Y × {1}. More generally, if {Xi | i ∈ I} is a family of topological spaces their
topological sum +i Xi is ∪i Xi × {i} with the topology making each Xj × {j}
clopen and each inclusion x 7→ (x, j) of Xj into +i Xi a homeomorphism to its
image Xj × {j}.
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Metric spaces again
Definition 9.1. A Cauchy sequence in a metric space X is a sequence of points
x1 , x2 , x3 , . . . in X such that, for every ε > 0, there exists N > 0 such that for
all n and m
m, n > N =⇒ dist(xm , xn ) < ε.
Definition 9.2. A metric space X is complete if every Cauchy sequence in X
converges in X.
Definition 9.3. A metric space X is totally bounded if, for every ε > 0, there
exists a finite cover of X by ε-balls.
Theorem 9.4. In a metric space X, the following are equivalent:
1. X is compact.
2. Each sequence in X has a convergent subsequence.
3. X is complete and totally bounded.
Definition 9.5. A Hausdorf space X is completely regular (or T3 21 ) if, for every
point x ∈ X and closed set C ⊆ X with x 6∈ C, there exists a map f : X → [0, 1]
such that f (0) = x and f ≡ 1 on C.
Proposition 9.6. Let X be a metric space. Define
(
1,
dist(x, y) > 1,
0
dist (x, y) =
dist(x, y), dist(x, y) ≤ 1.
Then dist0 and dist define the same topology on X.
Proposition 9.7. Let Xi , i = 1, 2, 3, . . . , be a metric space with metric bounded
P
i ,yi )
by 1. Define a metric on Πi Xi by dist(x, y) = i disti (x
, where xi is the ith
2i
coordinate of x, etc. Then this metric gives rise to the product topology.
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Lemma 9.8. Suppose that X is Hausdorff and that fi : X → [0, 1] are maps,
i = 1, 2, 3, . . . , such that, for every point x ∈ X and every closed set C ⊆ X
with x 6∈ C, there is an index i such that fi (x) = 0 and fi ≡ 1 on C. Define
f : X → Πi [0, 1] by f = Πi fi . Then f is an embedding, i.e., a homeomorphism
onto its image.
Lemma 9.9. Suppose that X is second countable and completely regular space
and let S be a countable basis for the open sets. For each pair U, V ∈ S with
U ⊆ V , select a map f : X → [0, 1] which is 0 on U and 1 on X − V , provided
such a function exists. Call this, possibly empty, set of maps F and note that
F is countable. Then, for each x ∈ X and each closed set C ⊆ X with x 6 inC,
there exists f ∈ F with f ≡ 0 on a neighborhood of x and f ≡ 1 on C.
Theorem 9.10 (Urysohn Metrization Theorem). If a space is second countable
and completely regular, then it is metrizable.
Lemma 9.11 (Lebesgue Lemma). Let X be a compact metric space and let
{Ui }i∈I be an open covering of X. Then, there exists δ > 0 (a Lebesgue number
for the covering) such that if A ⊆ X with diam(A) < δ then A ⊆ Uj , for some
j ∈ I.
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Existence of real valued functions
Lemma 10.1. Suppose that, on a topological space X, we are given, for each
dyadic rational number r in I (thus r = m/2n ), 0 ≤ m ≤ 2n ), an open set Ur
in X such that r < s =⇒ U r ⊆ U s . Then the function f : X → R defined by
(
inf{r | x ∈ Ur }, x ∈ U1
f (x) =
1,
x 6∈ U1
is continuous.
Lemma 10.2 (Urysohn’s Lemma). If X is normal and F ⊆ U , where F is
closed and U is open, then there is a map f : X → [0, 1] which is 0 on F and 1
on X − U .
Corollary 10.3. Normality implies complete regularity.
Corollary 10.4 (Tietze Extension Theorem). Let X be normal, F ⊆ X be
closed, and f : F → R be continuous. Then there is a map g : X → R, such
that g(x) = f (x) for all x ∈ F . Moreover, it can be arranged that
inf f (x) = inf g(x)
x∈F
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x∈R
and
sup f (x) = sup g(x).
x∈F
x∈R
Locally compact spaces
Definition 11.1. A topoogical space is locally compact if every point has a
compact neighborhood.
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Theorem 11.2. If X is a locally compact Hausodrff space then each neighborhood of a point x ∈ X contains a compact neighborhood of x (compact neighborhoods form a neighborhood basis at x). In particular, X is completely regular.
Theorem 11.3 (One-point compactification). Let X be a locally compact Hausodrff space. Put X + = X ∪ {∞}, where ∞ just represents some point not in
X. Define an open set in X + to be either an open set in X or a set of the form
X + − C, where C ⊆ is a compact. Then this defines a topology on X + which
makes X + into a compact Hausdorff space called the one-point compactification
of X. Moreover, this topology on X + is the only topology making X + compact
Hausdorff space with X as a subspace.
Theorem 11.4. Suppose that X and Y are locally compact, Hausdorff spaces
and that f : X → Y is continuous. Then f is proper if and only if f extends to
a continuous function f + : X + → Y + by setting f (∞X ) = ∞Y .
Proposition 11.5. If f : X → Y is a proper map between locally compact
Hausdorff spaces then f is closed.
Definition 11.6. A subspace A ⊆ X of a topological space is locally closed if
each point a ∈ A has an open neighborhood Ua in X such that Ua ∩ A is closed
in Ua .
Proposition 11.7. A subspace A ⊆ X is locally closed if and only if it has the
form A = C ∩ U , where C is closed and U is open in X.
Theorem 11.8. For a Hausdorff space the following are equivalent.
1. X is locally compact.
2. X is a locally closed subspace of a compact Hausdorff space.
3. X is a locally closed subspace of a locally compact Hausdorff space.
Definition 11.9. If X is a completely regular space, consider the set F of all
maps f : X → [0, 1]. Define
Φ : X → [0, 1]F
by
Φ(x)(f ) = f (x).
The closure of Φ(X) is called the Stone-Čech compactification of X and is
denoted β(X).
Theorem 11.10 (Stone-Čech compactification). If X is a completely regular
space, then β(X) is comapct Hausdorff and Φ(: X → β(X) is an embedding.
Theorem 11.11. If X is a completely regular space and f : X → R is a
bounded real valued map, then f can be extended uniquely to a map β(X) → R.
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Paracompact spaces
Definition 12.1. If U and V are open coverings of a space then U is said to
be a refinement of V if each element of U is a subset of some element in V .
Definition 12.2. A collection U of subsets of a topological space X is locally
finite if each point x ∈ X has a neighborhood N which intersects, nontrivially,
only a finite number of elements of U .
Definition 12.3. A Hausdorff space X is paracompact if every open covering
of X has an open, locally finite refinement.
Proposition 12.4. A closed subspace of a paracompact space is paracompact.
Theorem 12.5. A paracompact space is normal.
Definition 12.6. If f is a real valued map then the support of f is
{x | f (x) 6= 0}.
Definition 12.7. Let {Ui | i ∈ I} be an open covering of the space X. Then a
partition of unity subordinate to this covering is a collection of maps
{ fj : X → [0, 1] | j ∈ J }
such that:
1. There is a locally finite open refinement {Vj | j ∈ J} such that the support
of fj is in Vj for all j ∈ J; and
P
2.
j fj (x) = 1 for each x ∈ X.
Theorem 12.8. If X is a paracompact and U is an open covering of X then
there exists a partition of unity subordinate to U .
Proposition 12.9. If x is paracompact and {Ui } is a locally finite open covering
of X then there exists an open covering {Vi } such that V j ⊆ Uj , for j ∈ J.
Definition 12.10. A space is called σ-compact if it is the union of countably
many compact subspaces.
Definition 12.11. A locally compact Hausdorff space is paracompact if and
only if it is the disjoint union of open σ-compact subsets.
Theorem 12.12. If X is locally compact, Hausdorff, and second countable,
then its one-point compactification is metrizable and X is σ-compact and paracompact.
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Quotient spaces
Definition 13.1. Let X be a topological space, Y a set, and f : X → Y an
onto function. Them we define a topology on Y , called quotient topology or
topology induced by f , by specifying a set V ⊆ Y to be open if and only if
f −1 (V ) is open in X. Note that this is the largest topology on Y that makes f
continuous.
Definition 13.2. Let X be a topological space and ∼ an equivalence relation
on X. Let Y = X/ ∼ be the set of equivalence classes and π : X → Y the
canonical function taking x ∈ X to its equivalence class [x] ∈ Y . Then, Y with
the topology induced by π is called a quotient space of X.
Proposition 13.3. A quotient space of a quotient space of X is a quotient
space of X. That is, if X → Y toZ are two onto functions and Y is given the
quotient topology from X, and Z the quotient topology from Y , then Z has the
quotient topology from X induced by the composition of the two functions.
Definition 13.4. A map X → Y is called an identification map if it is onto
and Y has the quotient topology.
Definition 13.5. A surjection f : X → Y is an identification map if and only
if, for all functions g : Y → Z, g is continuous if and only if g ◦ f is continuous.
Example 13.6 (Projective plane). P2 = D2 /(∼) = S2 /(∼).
Definition 13.7. If X is a space and A ⊆ X, then X/A denotes the quotient
space obtained via the equivalence relation whose equivalence classes are A and
the singletons {x}, for x ∈ X − A.
Proposition 13.8. If X is regular and A is closed, then X/A is Hausdorff. If
X is normal and A is closed, then X/A is normal.
Example 13.9 (Disks from cylinders). Dn+1 = (Sn × I)/(Sn × {0}).
Example 13.10 (Spheres from disks). Sn = Dn /Sn−1 .
Definition 13.11. If A ⊆ X and ∼ is an equivalence relation on X then the
saturation of A is {x ∈ X | (∃a ∈ A) x ∼ a} (thus, it is ∪a∈A [a]).
Proposition 13.12. If A ⊆ X and ∼ is an equivalence relation on X such
that every equivalence class intersects A nontrivially then the induced map k :
A/ ∼→ X/ ∼ is a homeomorphism if the saturation of every open (respectively,
closed) set of A is open (respectively, closed) in X.
Definition 13.13 (Attaching). Let X and Y be spaces, A ⊆ X be closed and
f : A → Y be a map. We denote by Y ∪f X the quotient space of the disjoint
union Y + X by the equivalence relation generated by the relation a ∼ f (a), for
a ∈ A.
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Proposition 13.14. The canonical map Y → Y ∪f X is an embedding into a
closed subspace. The canonical embedding X − A → Y ∪f X is an embedding
into on open subspace.
Definition 13.15. If X → Y is a map then the mapping cylinder of f is the
space Mf = Y ∪f0 X × I, where f0 : X × {0} → Y is f (x, 0) = f (x), for all
x ∈ X.
Definition 13.16. If X → Y is a map then the mapping cone of f is the space
Cf = Mf /(X × {1}).
Proposition 13.17. A function Mf → Z is continuous if and only if the
induced functions X × I → Z and Y → Z are both continuous.
Proposition 13.18. If f : X → Y is an identification map and K is a locally
compact Hausdorff space then f × 1 : X × K → Y × K is an identification map.
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Homotopy
Definition 14.1. If X and Y are topological spaces a homotopy of maps from
X to Y is a map F : X × I → Y .
Two maps f0 , f1 : X → Y are homotopic if there exists a homotopy F :
X ×I → Y of maps from X to Y such that F (x, 0) = f0 (x) and F (x, 1) = f1 (x),
for x ∈ X. In this case, we write f0 ' f1 .
Proposition 14.2. If f, g : X → Y h : X 0 → X and kY → Y 0 , then
f 'g
=⇒
f ◦h'f ◦f
and
k ◦ f ' k ◦ g.
Definition 14.3. A map f : X → Y is homotopy equivalence with homotopy
inverse g if there is a map g : Y → X such that g ◦ f ' 1X and f ◦ g ' 1y .
This relation is denoted by X ' Y . In this case one says that X and Y have
the same homotopy type.
Definition 14.4. A space is contractible if it is homotopy equivalent to the
one-point space.
Proposition 14.5. A space X is contractible if and only if the identity map
1X is homotopic to a constant map r : X → X.
Example 14.6. Rn is contractible.
Example 14.7. Sn−1 ' Rn − {0}.
Definition 14.8. A subspace A ⊆ X is strong deformation retract of X if there
exists a homotpy F : X × I → X (called deformation) such that:
F (x, 0) = x, for x ∈ X
F (x, 1) ∈ A, for x ∈ X
F (a, t) = a, for a ∈ A, t ∈ I.
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Example 14.9. If f : X → Y is a map then the canonical map r : Mf → Y is
a strong deformation retraction.
Definition 14.10. If A ⊆ X then a homotopy F : X × I → Y is relative to A
(or rel A) if F (a, t) is independent of t, for a ∈ A. A homotopy that is rel X is
called constant homotopy.
Definition 14.11. If F : X × I → Y and G : H × I → Y are homotopies such
that F (x, 1) = G(x, 0), for x ∈ X, then define a homotopy F ∗ G : X × I → Y ,
called concatenation of F an G, by
(
F (x, 2t),
t ≤ 1/2,
(F ∗ G)(x, t) =
G(x, 2t − 1), t ≥ 1/2.
Lemma 14.12 (Reparametrization Lemma). Let φ1 , φ2 : (I, ∂I) → (I, ∂I) be
two maps which agree on ∂I. Let F : X × I → Y be a homotopy and let
Gi (x, t) = F (x, φi (t)), for i = 1, 2. Then G1 ' G2 rel X × ∂I
Proposition 14.13. We have F ∗ C ' F rel X × ∂I and, similarly, C ∗ F '
F rel X × ∂I.
Definition 14.14. If F : X × I → Y is a homotopy, then we define F −1 :
X × I → Y by F −1 (x, t) = F (x, 1 − t).
Proposition 14.15. For a homotopy F we have F ∗F −1 ' C rel X ×∂I, where
X(x, t) = F (x, 0) for all x and t, i.e., C is a constant homotopy.
Proposition 14.16. For any homotopies F , G, and H for which the concatenations F ∗G and G∗H are defined, we have F ∗(G∗H) ' (F ∗G)∗H rel X ×∂I.
Proposition 14.17. For homotopies F1 , F2 , G1 and G2 , if F1 ' F2 rel X × δI
and G1 ' G2 rel X × δI, then F1 ∗ G1 ' F2 ∗ G2 rel X × δI.
Theorem 14.18. If f0 ' f1 : X → Y then Mf0 ' Mf1 rel Y + X and Cf0 '
Cf1 rel Y + vertex.
Theorem 14.19. If φ : Y → Y 0 is a homotopy equivalence then so is F :
(Mf , X) → (Mφ◦f , X) and hence so is F : Cf → Cφ◦f .
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Topological groups
Definition 15.1. A topological group is a Hausdorff topological space G together with a group stucture on G such that:
1. group multiplication (g, h) 7→ gh from G × G → G is continuous, and
2. group inversion g 7→ g −1 from G → G is continuous.
Definition 15.2. A subgroup of a topological group is a subspace which is also
a subgroup in the algebraic sense.
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Definition 15.3. If G and G0 are topological groups then a homomorphism
f : G → G0 is a group homomorphisms which is also continuous.
Definition 15.4. If G is a topological group and g ∈ G, then a left translation
by g is the map Lg : G → G given by L(h) = gh. Similarly, the right translation
by g is the map Rg : G → G given by Rg (h) = hg −1 .
Proposition 15.5. In a topological group G we have Lg ◦ Lh = Lgh and
Rg ◦ Rh = Rgh . Moreover, both Lg and Rg are homeomorphisms, as is the
conjugation by g (given by h 7→ ghg −1 ) and inversion (h 7→ h−1 ).
Definition 15.6. A subset A of a topological group is called symmetric if
A = A−1 .
Proposition 15.7. In a topological group G with unity element e, the symmetric neighborhoods of e form a neighborhood basis at e.
Proposition 15.8. If G is a topological group, g ∈ G, and U is any neighborhood of g, then there exists a symmetric neighborhood V of e such that
V gV −1 ⊆ U .
Proposition 15.9. If G is a topological group, U is any neighborhood of e, and
n is any positive integer, then there exists a symmetric neighborhood V of e such
that V n ⊆ U .
Proposition 15.10. If H is a subgroup of a topological group G then H is also
a subgroup of G. If H is normal subgroup of G then so is H.
Proposition 15.11. If G is a topological group G and H is a closed subgroup,
then the space G/H of left cosets of H in G, with the topology induced by the
canonical function π : G → G/H, is a Hausdorff space. Moreover, π is open.
Proposition 15.12. If H is a closed, normal subgroup of the topological group
G then G/H, with the quotient topology, is a topological group.
Definition 15.13. If G is a topological group and X is a space, then an action
of G on X is a map G × X → X, with the image of (g, x) being denoted by
g(x), such that, for g, h ∈ G and x ∈ X:
1. (gh)(x) = g(h(x)), and
2. e(x) = x
Proposition 15.14. If G is a compact topological group acting on the Hausdorff
space X and Gx is the isotropy group at x, then the map φ : G/Gx → G(x)
given by gGx = g(x) is a homeomorphism.
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Convex bodies
Definition 16.1. A convex body in Rn is a closed set C ⊆ Rn with the property
that, whenever p, q ∈ C, the line segment between p and q is contained ion C.
Proposition 16.2. If C ⊆ Rn is a convex body and 0 ∈ int(C), then any ray
from the origin intersects ∂C at at most one point.
Proposition 16.3. Let C ⊆ Rn be a compact, convex body with 0 ∈ int(C).
x
then the function f : ∂C → Sn−1 , given by f (x) = ||x||
, is a homeomorphism.
Theorem 16.4. A compact convex body C in Rn with nonempty interior is
homeomorphic to the closed n-ball Dn and ∂C ≈ Sn−1 .
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The Baire Category Theorem
Theorem 17.1 (Baire Category Theorem). Let X be either a complete metric
space or a locally compact Hausdorff space. Then the union of a countably many
nowhere dense subsets of X has empty interior.
Definition 17.2. A subset S of a space X is of first category if it is the countable
union of nowhere dense subsets. Otherwise it is of second category. A set of
second category is said to be residual if its complement is a set of first category.
Corollary 17.3. Let X be either a complete metric space or a locally compact
Hausdorff space. Then the intersection of any countable family of dense open
sets in X is dense.
Corollary 17.4. If {fn } is a sequence of continuous functions fn : X → Y
from a complete metric space X to a metric space Y and if lim fn (x) = f (x)
exists for each x ∈ X, then the set of points of continuity of f is residual and
hence dense.
Corollary 17.5. There exists a connected 2-manifold (i.e., a Hausdorff space
in which each point has a neighborhood homeomorphic to the plane with the
following properties:
1. it has a countable dense subset.
2. it has an uncountable discrete subset, and hence is not second countable
3. it is not normal, and hence not metrizable.
Corollary 17.6. In the space RI of continuous finctions f : I → R in the
uniform metric, the set of functions which are nowhere differentiable is dense.
Indeed, it is residual in RI .
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