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MA4266 Topology

Lecture 15. Tuesday 30 March 2010. MA4266 Topology. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@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. Chains and Maximal Elements. Assume that.

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MA4266 Topology

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  1. Lecture 15. Tuesday 30 March 2010 MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@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

  2. Chains and Maximal Elements Assume that is a partially ordered set. Definition A subset is called a chain if is a linearly (or totally) ordered set. Definition An element is a maximal element if Remark If has a maximum then is a maximal element. However a maximal element need not be a maximum element. Example are chains, are not chains, are (all 3) maximal elements.

  3. Zorn’s Lemma Lemma (Zorn) Assume that is a partially ordered set. If every chain has an upper bound then has a maximal element. Proof This is equivalent to the axiom of choice, the well ordering principle, and the Hausdorff maximal principle. http://en.wikipedia.org/wiki/Zorn%27s_lemma Lemma Hausdorff Maximal Principle (HMP) Every chain in is contained as a subset of a maximal chain. Proof It suffices by Zorn’s lemma to show that every chain of chains has a maximal element. This follows since the union of a chain of chains is itself a chain which is also an upper bound for the chains in the chain.

  4. Alexander Subbasis Theorem Theorem A space is compact if and only if there exists a subbasis for with the following Property F: Every cover of by members of has a finite subcover. Proof If is compact then every subbasis has prop. F. To prove the converse assume that is a subbasis with property F and assume that is not compact. It suffices to obtain a contradiction. Let be an open cover of that has no finite subcover. Covers are partially ordered by inclusion and the union of any chain of covers, each having no finite subcover, is an upper bound of the covers in this chain. So the HMP implies

  5. Alexander Subbasis Theorem that there exists a maximal open cover of with the property that it has no finite subcover. So we can and will choose to be a maximal open cover with this property. (remark: this required the Axiom of Choice) Lemma If are open sets and there exists such that then there Proof Else exists such that such that This gives the contradiction

  6. Alexander Subbasis Theorem there exist Now for every and every such that Question Why does this statement hold ? Then the preceding lemma implies that there exists Therefore such that is an open cover of by members of and hence admits a finite subcover Since it follows that is a finite subcover of by members of This contradiction completes the proof.

  7. The Tychonoff Theorem Theorem7.11The product of compact spaces is compact Proof Let be a collection of compact be the product space and spaces, let be the let ‘standard’ subbasis. It suffices, by the AST to show that every cover of by elements of has a finite subcover. Let be such a cover and assume to the contrary that did not have a finite subcover. For each let No finite subset of covers so doesn’t cover so that is not in any member of then If

  8. Assignment 15 Read pages 234-237, 237-241, 243-251. Written Homework #3 Due Friday 9 April 1. Let be a nonempty set. A subset is called ‘frisbee’ (on X) if it satisfies the following: is called a ‘freeflyer’ if and is called a ‘highflyer’ it is a frisbee that is not a proper subset of another frisbee.

  9. Assignment 15 (a) For nonempty define Prove that is always a frisbee and that it is a a highflyer if and only if S is a singleton set. (b) For show that is a freeflyer. (c) Use the Hausdorff Maximal Principle to prove that every frisbee is contained in a highflyer. (d) Show that a frisbee is a highflyer if and only if for every either or Remark (b) and (c) ensure the existence of freeflying highflyers, and (d) shows that they are mind boggling !

  10. Assignment 15 2. Let be a nonempty topological space and A frisbee (on X) ‘lands on’ if every open set that contains belongs to (a) Let and be topological spaces. Show that a function is continuous at if and only if for every frisbee (on X) that lands on (clearly a frisbee on Y) lands on (b) Let be a product space and let be a frisbee on X. Prove that lands on if and only if lands on

  11. Assignment 15 (c) Prove that a space is compact if and only if every highflyer lands on some point in (d) Prove The Tychonoff Theorem using highflyers.

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