CALTECH MA 109A - Lecture notes (4 pages)

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Lecture notes



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Lecture notes

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Pages:
4
School:
California Institute of Technology
Course:
Ma 109a - Introduction to Geometry and Topology
Introduction to Geometry and Topology Documents

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From Urysohn s lemma we can deduce that for normal spaces continuous functions to 0 1 separate points Proposition 5 36 In a normal space X for any two points x y X x 6 y we can find continuous f X 0 1 with f x 6 f y Urysohn s lemma makes it possible to find a function on a space that roughly approximates a function on a subset Lemma 5 37 Let X be normal and let A X be closed Suppose f A 1 1 is continuous Then there is a continuous function g X 1 3 1 3 such that 2 sup f x g x 3 x A Proof Let B f 1 1 3 1 and let C f 1 1 1 3 By Urysohn s lemma there is a function on X that is 1 on B and 0 on C by rescaling this function we get the desired function g This lemma is key to the following theorem Theorem 5 38 Tietze extension theorem A space X is T4 if and only if for every closed subset A and every continuous function f A R there is a function g X R extending f meaning g A f Proof sketch A rigorous proof of this theorem requires quoting some basic results from analysis so we give a sketch instead The if direction is easy and is omitted For the only if direction we compose f with a homeomorphism R 1 1 to get a bounded function Assuming f is bounded we approximate f by a function f1 defined on X using the lemma The difference is presumably nonzero so we approximate the difference by another function f2 and then f1 f2 is a function on X that restricts to a better approximation of f Repeating this process infinitely many times the error of the approximations goes to zero exponentially fast and the approximations converge to the desired function g this uses the fact from analysis that uniform limits of continuous functions are continuous More on metric spaces Some last comments about separation and countability properties as they pertain to metric spaces As we mentioned before metric spaces are separable if and only if they are second countable However taking the balls of rational radius around each point gives us the following 27 Proposition 5 39 Every metric space is first



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