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M21–651, Introduction to Topology Professor Walkington 1 Weak Topologies Definition 1.1 Let X be a set and {Xα }α∈A be a family of topological spaces and fα : X → Xα . The weak topology on X induces by {fα | α ∈ A} is the smallest topology on X for which each fα is continuous. The following lemma summarizes basic properties of the weak topology. Lemma 1.2 Let X have the weak topology induced by a collection {fα : X → Xα | α ∈ A}. 1. Sets of the form fα−1 (Uα ) with Uα sub-basic open sets in Xα form a subbase for the topology on X. 2. If Y is a topological space then f : Y → X is continuous if and only if fα ◦ f : Y → Xα is continuous. 3. If B ⊂ X, then the subspace topology is induced by the restriction of fα to B. 4. If each Xα has the weak topology induced by a collection {gαλ : Xα → Yαλ | λ ∈ Λα } then X has the weak topology given by the collection {gαλ ◦ fα : X → Yαλ | λ ∈ Λα }. Proof. 1. A sub-base for the weak topology is S = {fα−1 (Uα ) | Uα ⊂ Xα is open}. If Sα is a sub-base for Xα , then any open set Uα in Xα may be written as Uα = ∪i∈I ∩nj=1 Sαij 1 with Sαij ∈ Sα , for some index set I and integer n. Then fα−1 (Uα ) = ∪i∈I ∩nj=1 fα−1 (Sαij ), which implies S 0 = {fα−1 (Sα ) | Sα ∈ Sα } is a also a sub-base for the weak topology. 2. This is a crucial property. If f is continuous, then so too is the composition fα ◦ f : Y → Xα . Next, suppose each of these functions is continuous. Then a sub-basic open set in X takes the form S = fα−1 (Uα ) with Uα ⊂ Xα open. Then f −1 (S) = (f ◦ fα )−1 (Uα ), being open by hypothesis, shows that the inverse image of a sub-basic open set is open, so f is continuous. 3. Let Uα ⊂ Xα be open, then (fα |B )−1 (α) = fα (Uα ) ∩ B. The set on the left is a sub-basic open set for the weak topology on B, and the set on the right is a sub-basic open set for the sub-space topology; hence the two topologies agree. 4. A sub-basic open set for the topology on Xα takes the from Uα = −1 gαλ (Vαλ ), where Vαλ ⊂ Yαλ is open. Then a sub-basic open set in X takes the from fα−1 (Uα ) = (gαλ ◦ fα )−1 (Vαλ ). Definition 1.3 Let fα : X → Xα , for each α ∈ A. 1. The evaluation map e : X → Q α Xα is defined by e(x)α = fα (x). 2. The collection {fα | α ∈ A} separates points in X if whenever x 6= y in X there exists α ∈ A such that fα (x) 6= fα (y). 2 3. The collection {fα | α ∈ A} separates points from closed sets in X if whenever B is closed in X and x 6∈ B there exists α ∈ A such that fα (x) 6∈ fα (B). Recall that the continuous functions from a completely regular space X to [0, 1] separate points from closed sets, and Urysohn’s theorem shows that all T1 normal spaces are completely regular. Definition 1.4 Let X and Y be topological spaces and f : X → Y be injective and continuous and let Z = f (X) have the subspace topology. If f : X → Z is a homeomorphism then the mapping f : X → Y is a (topological) embedding. The following theorem is used ubiquitously to characterize various topological spaces as subspaces of a product of typically simpler spaces (e.g. Xα = [0, 1] for all α). Theorem 1.5 Let X and Xα be topological spaces for each α ∈ A and fα : Q X → Xα . Then the evaluation map e : X → α Xα is an embedding if and only if X has the weak topology induced by the collection {fα | α ∈ A} and this collection separates points. Proof. (⇒) This is the easy half. An embedding separates points, thus if x 6= y it follows that e(x) 6= e(y) so for some α fα (x) = πα ◦ e(x) 6= πα ◦ e(y) = fα (y). Next, a sub-basic open subset of e(X) in the subspace topology of the product takes the from Sα = πα−1 (Uα ) ∩ e(X), with Uα ⊂ Xα open. Since e is an embedding it follows S = e−1 (Sα ) is a sub-basic open set for X. Then S = e−1 (Sα ) = e−1 (πα−1 (Uα )) = (πα ◦ e)−1 (Uα ) = fα−1 (Uα ), shows that S is a sub-basic open set of the weak topology on X. thus the topology on X is no coarser than the weak topology. Moreover, since fα = πα ◦ e is a composition of continuous functions each fα is continuous, so the topology on X contains the weak topology induced by the fα . 3 Q (⇐) Since the embedding separates points it follows that e : X → α Xα is injective. Also, since πα ◦ e = fα and the latter is continuous when X has the weak topology, it follows that πα ◦ e is continuous for all α which implies e is continuous. It then suffices to show that e is an open map. Let U = fα−1 (Uα ) be a sub-basic open set in X. Since e is a bijection we have πα−1 (Uα ) ∩ e(X) = = = = e ◦ e−1 ◦ πα−1 (Uα ) e ◦ (πα ◦ e)−1 (Uα ) e ◦ fα−1 (Uα ) e(U ). The set on the left is a sub-basic open set of the subspace e(X). If X is a topological space the following lemma can be used to determine if it is the weak topology of a collection {fα | α ∈ A} Lemma 1.6 A collection {fα : X → Xα | α ∈ A} of continuous functions on a topological space X separates points from closed sets in X if and only if the sets fα−1 (V ), for α ∈ A and V open in Xα , form a base for the topology on X. Proof. (⇒) Fix U ⊂ X open, and for each x ∈ U let fα (x) be disjoint from Bx ≡ fα (X \ U ). Then x ∈ fα−1 (Xα \ Bx ) = X \ fα−1 (Bx ) ⊂ U. It follows that U = ∪x∈U fα−1 (Xα \ Bx ). (⇐) Let the sets form a base for the topology, and B ⊂ X be closed and x 6∈ B. Then x ∈ X \ B is open, so there exists a basic open set such that x ∈ fα−1 (V ) ⊂ X \ B, V ⊂ Xα open. Then f (B) ∈ X \ V and the set on the right is closed, so fα (B) ⊂ X \ V , and since fα (x) ∈ V it follows f (x) 6∈ fα (B) Corollary 1.7 If {fα : X → Xα | α ∈ A} is a collection of continuous functions on a topological space X which separates points from closed sets, then the topology on X is the weak topology induced by the fα . 4 Recall that a topological space is T1 if and only if points are closed. Theorem 1.8 If X is a T1 space and {fα : X → Xα | α ∈ A} is a collection of continuous functions Q which separates points from closed sets, then the evaluation e : X → α Xα is an embedding. Proof. The corollary shows that X has the weak topology induced by the fα , and since X is a T1 space, a collection which separates points from closed sets also separates of points. Theorem ?? then shows that the evaluation map is an embedding. 5