1. MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sgmatwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Lecture 13

2. Finite Products Example 7.1.1 Question Why are the projections maps continuous ? with the usual topology has a basis the set Question Isa topology for Question What is the ‘usual’ basis for Question How can this example be generalized to define a basis for a topology on the product of spaces

3. Finite Products Theorem 7.1 Letwhereis a space and is a product spaceThenis continuous iff the composition is continuous for each projection Proof First observe that a subbasis foris given by Theorem 4.11 (p 116) implies thatis continuous iff eachis open.

4. Finite Products Theorem 7.2 The product of a finite number of Hausdorff spaces is a Hausdorff space. Theorem 7.3 The product of a finite number of connected spaces is a connected space. Theorem 7.4 The product of a finite number of separable spaces is separable. Theorem 7.5 The product of a finite number of 1 st (2 nd ) countable spaces is 1st (2nd) countable. Theorem 7.6 Ifthen the product topology onis the topology generated by the product metric defined on page 83.

5. Finite Products Lemma In order that a spacebe compact it is sufficient that there exist a basisfor every cover such that Proof Letbe such a basis and let open cover of be an For eachthere exists with has a finite subcover. andwith Then is an open cover ofso it has a finite subcover hencecovers of

6. Finite Products Theorem 7.7 The product of a finite number of compact spaces is compact. Proof It suffices to show that ifand compact then are is compact. Let be the basis for the product topology and be a cover of Forthe subset is compact so it is a subset of where eachLetThen Read rest of proof p. 202-203.

7. Arbitrary Products Definition Ifis an index set and a family of sets, their Cartesian product is the set: For the function given byis called the projection map ofonto thecoordinate set Definition If eachis a nonempty topological space, the product topology onis the topology generated by the subbasis

8. Arbitrary Products Theorem 7.8 Generalization of Theorem 7.1 that characterizes continuity of maps using projections. Theorem 7.9 Hausdorff space products are Hausdorff Theorem 7.10 Connected space products are conn. Proof Letand let be the subset ofof points that differ from in at mostcoordinates.Problem: Show that each connected. Thenis connected andthereforeis connected.

9. Arbitrary Products Lemma: The Alexander Subbasis Theorem A space is compact iff there exists a subbasis Proof See exercise 12 on p. 210-211. for such that every coverofhas a finite subcover. Theorem 7.11: The Tychonoff Theorem The product of compact spaces is compact. Proof: Letcover We show thathas a finite subcover. For each letbe the collection of opensuch that Problem: prove thatdoes not cover Constructso that for every Thencontradictingcovering

10. Examples Example 7.2.1 The Hilbert Cube is the product of a countably infinite number of closed intervals. It is embedded in Hilbert space can be embedded in the Hilbert Cube by the map by the map Infinite Dimensional Euclidean Space is homeomorphic toA deeper result: Proof. R. D. Anderson and R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 74 (1968),771-792.

11. Examples Example 7.2.2 The Cantor Set is an Infinite Product The Cantor set, consisting of all numbers in having the ternary (base 3) expansion is homeomorphic to the infinite product by the map Proof See page 209.

12. Assignment 13 Read pages 195-203, 204-209 Prepare to solve during Tutorial Exercise 7.1 problems 4, 10, 11 Exercise 7.2 problems 5, 11, 12