Download p. 1 Math 490 Notes 11 Initial Topologies: Subspaces and Products

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Math 490 Notes 11
Initial Topologies: Subspaces and Products
We’ve already visited (briefly) subspace and finite product space topologies, as presented in
Munkres. Here, we’ll re-visit these topics from a more general point of view. We concern ourselves with such topics, in part, because many interesting and important topological spaces
are constructed from other well-known spaces, such as (R, τu ), by forming subspaces, product spaces, quotient spaces, and disjoint sums, or various combinations of these. Subspace
and product topologies are special cases of initial topologies (defined below); quotient and
disjoint sum topologies are special cases of final topologies, which we may discuss later.
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Def. N11.1 Let X be a set, let {(Xi , τi ) ¯ i ∈ I} be a collection of topological spaces, and
¯
let {fi ¯ i ∈ I} be a corresponding collection of functions, where fi : X → Xi . The topology
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τ on X with subbasis S = {fi−1 (Ui ) ¯ Ui ∈ τi , i ∈ I} is called the initial topology on X
¯
¯
generated by {fi ¯ i ∈ I} and {(Xi , τi ) ¯ i ∈ I}.
Note that τ has a basis of sets of the form
n
\
j=1
¡ ¢
Uij , where {i1 , i2 , . . . , in } ⊆ I and
fi−1
j
Uij ∈ τij for i = 1, 2, . . . , n; we’ll refer to these as basic open sets for τ .
Theorem N11.1 In the notation of the previous definition, τ is the coarsest topology on X
relative to which fi : (X, τ ) → (Xi , τi ) is continuous for all i ∈ I.
Proof : To show that fi is continuous for every i, note that Ui ∈ τi ⇒ f −1 (Ui ) ∈ τ by
definition.
To show that τ is the coarsest such topology, assume µ is a topology on X such that
fi : (X, µ) → (Xi , τi ) is continuous for all i ∈ I. We must show that τ ≤ µ. But this follows
immediately from the fact that each subbasic set fi−1 (Ui ) in τ is also in µ, by the assumed
continuity of fi : (X, µ) → (Xi , τi ).
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Topological Subspaces
Def. N11.2 Let (X1 , τ1 ) be a topological space, and let φ 6= X ⊆ X1 . Let iX : X → X1 be
the identity injection. The initial topology τ on X induced by the single function iX and
(X1 , τ1 ) is called the subspace topology on X, and (X, τ ) is called a topological subspace
of (X1 , τ1 ).
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¯
Note that since inverse images preserve unions and intersections, we have τ = {i−1
X (V ) V ∈ τ1 },
¯
¯ V ∈ τ1 }, which agrees with the definition
and i−1
(V
)
=
V
∩
X,
so
we
have
τ
=
{V
∩
X
X
of the subspace topology in Munkres, given in Notes 7. If (X, τ ) is a topological space and
A ⊆ X, then the subspace topology on A is commonly denoted by τA .
Example 1
Consider X = [0, 1] as a subspace of (R, τu ). Note that sets of the form
[0, ǫ) and (1 − ǫ, 1] (for 0 < ǫ ≤ 1) are open in the subspace topology, but are not τu -open.
As we’ve mentioned before, in this particular case, the subspace topology agrees with the
order topology on [0, 1].
Example 2 Consider X = [0, 1) ∪ {2} as a subspace of (R, τu ). Note that {2} = ( 32 , 25 ) ∩ A
is an isolated point in the subspace topology, but not in the order topology on A.
Prop. N11.1
(a) If (A, τA ) is a subspace of (X, τ ) and B ⊆ A, then B is τA -closed iff ∃ a τ -closed set C
such that B = A ∩ C.
(b) If D ⊆ A, then ClτA D = (Clτ D) ∩ A.
Proof :
(a) This is basically a consequence of DeMorgan’s laws. B is τA -closed iff A − B is τA -open,
which is true iff there is a τ -open set Ĉ such that A − B = Ĉ ∩ A. But in this case, X − Ĉ
is τ -closed, and (X − Ĉ) ∩ A = B.
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(b) By (a), Clτ D ∩ A is a τA -closed overset of D, and so ClτA D ⊆ (Clτ D) ∩ A.
To prove the opposite inclusion, suppose x ∈ (Clτ D) ∩ A. If U is any τA -nbhd of x, then
U = V ∩ A, where V is a τ -nbhd of x. By definition of closure, we must have V ∩ D 6= φ,
which implies (since D ⊆ A) that (V ∩ A) ∩ D = U ∩ D 6= φ. Thus x ∈ ClτA D.
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Topological Products
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Recall our definition for set product: If {Xi ¯ i ∈ I} is an arbitrary collection of sets, then
their product set is
Y
[ ¯¯
¯
X=
Xi = {f : I →
Xi ¯ f (i) ∈ Xi for all i ∈ I} = {(xi ) ¯ xi ∈ Xi for all i ∈ I}.
i∈I
i∈I
In the latter representation, we write f = (xi ) iff f (i) = xi for all i ∈ I. This is consistent
with the usual ”n-tuple” notation for finite products:
n
Y
¯
Xi = X1 × X2 × . . . × Xn = {(x1 , x2 , . . . , xn ) ¯ xi ∈ Xi for i = 1, 2, . . . , n}.
i=1
Q
For any product set X = i∈I Xi , the j-th projection map pj : X → Xj is defined by
pj (f ) = f (j), or in the more informal notation, pj ((xi )) = xj . That is, the projection map
pj maps each element in the product set to its ”j-th component”. Munkres denotes the jth
projection map pj by πj .
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Def. N11.3 Let {Xi ¯ i ∈ I} be an arbitrary collection of topological spaces. We define
Y
the product topology τ on X =
Xi to be the initial topology induced by the given
i∈I
¯
collection of topological spaces, together with the projection maps {pj ¯ i ∈ I}. In this case,
Q
we write (X, τ ) = i∈I (Xi , τi ).
¯
¯
Note that τ has as a subbasis S = {p−1
i (Ui ) Ui ∈ τi , i ∈ I}, and a basic set for τ is any
finite intersection of members of S. By Thm N11.1, the product topology is the coarsest
topology on the product set X which makes all the projection maps pi : X → Xi continuous.
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If I = {1, 2, . . . , n} is finite, the basic open sets for τ are relatively easy to describe, as follows.
n
\
If Uj ∈ τj for j = 1, 2, . . . , n, then U =
p−1
j (Uj ) = U1 × U2 × . . . × Un is a basic τ -open
j=1
set. For example, the usual topology for R2 is the product topology for (R, τu ) × (R, τu ),
which has basic open sets of the form U1 × U2 , where U1 and U2 are open intervals in R. If
Q
X = i∈I Xi is an infinite product (meaning I is an infinite set), then a basic open set U for
n
\
the product topology has the form U =
p−1
ij (Uij ), where {i1 , i2 , . . . , in } ⊆ I and Uij ∈ τij
j=1
for j = 1, 2, . . . , n. Note that U is ”restricted” in only finitely many components. Another
representation for U is
U = (U1 × . . . × Un ) ×
Y
Xi .
i∈I−{1,...,n}
As is done in Munkres, the above expression is often taken as the definition of the basis
elements for the product topology on an arbitrary product space.
It might seem more natural to define basic open sets to be of the form
Q
Ui , where Ui ∈ τi
Q
for all i ∈ I. Indeed, such sets V do form a basis for a topology on X = i∈I Xi , and the
i∈I
topology obtained in this way is called the box topology, denoted τB . Cleary every open
set in the product topology τ is also τB -open so τ ≤ τB . The product and box topologies are
the same for finite products, but not for infinite products (unless all but finitely many of the
Xi ’s are indiscrete). When refering to ”product spaces”, we’ll always assume the product
topolgy, unless otherwise indicated.
Prop. N11.2 In a product space, any product of closed sets is closed.
Q
Q
Proof : Let (X, τ ) = i∈I (Xi , τi ), and let Ai ⊆ Xi be τi -closed for all i ∈ I. Let A = i∈I Ai .
To show A is closed, let x = (xi ) 6∈ A. Then there exists j ∈ I such that xj 6∈ Aj . Let
Uj = Xj − Aj . Then by assumption, Uj is τj -open, so V = p−1
j (Uj ) is τ -open (by continuity
of pj : (X, τ ) → (Xj , τj )). Finally, since x ∈ V ∈ τ and V ∩ A = φ, we’ve shown that X − A
is τ -open, and thus A is τ -closed.
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Example 3 For the Euclidean space Rn , a basis consists of all products of the form
(a1 , b1 ) × (a2 , b2 ) × · · · × (an , bn ).
Because Rn is a finite product, its box topology and product topology are equivalent.
Example 4 Consider the infinite product Rω . This set can be thought of as consisting
of all countably infinite sequences of real numbers of the form
(r1 , r2 , r3 , . . .).
Define the function f : R → Rω by
f (t) = (t, t, t, . . .).
Since the nth component function here is fn (t) = t, we have (for any collection {Ui } with
Ui ⊆ R)
f −1
¡Y
i∈Z+
\
¢
Ui .
Ui =
i∈Z+
A basis element for the product topology in Rω is a product of finitely many intervals (a, b)
with infinitely many copies of R. The inverse under f of such a set will be the intersection
of finitely many intervals in R, which is open in (R, τu ). So f is continuous from (R, τu ) to
Rω with the product topology.
On the other hand, the set
Y ¡ 1 1¢
¡ 1 1¢ ¡ 1 1¢
= (−1, 1) × − ,
× − ,
× ···
− ,
V =
i i
2 2
3 3
i∈Z+
is open in the box topology on Rω , but not in the product topology. Its inverse,
³ Y ¡ 1 1 ¢´
\ ¡ 1 1¢
−1
− ,
− ,
f
=
= {0}
i i
i i
i∈Z
i∈Z
+
+
is not open in (R, τu ). So f is not continuous from (R, τu ) to Rω with the box topology.