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S35 - 1st December 2004 Munkres 35 Ex 35.3 Let X be a...

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1st December 2004 Munkres § 35 Ex. 35.3. Let X be a metrizable topological space. (i) (ii): (We prove the contrapositive.) Let d be any metric on X and ϕ : X R be an unbounded real-valued function on X . Then d ( x, y ) = d ( x, y ) + | ϕ ( x ) - ϕ ( y ) | is an unbounded metric on X that induces the same topology as d since B d ( x, ε ) B d ( x, ε ) B d ( x, δ ) for any ε > 0 and any δ > 0 such that δ < 1 2 ε and d ( x, y ) < δ ⇒ | ϕ ( x ) - ϕ ( y ) | < 1 2 ε . (ii) (iii): (We prove the contrapositive.) Let X be a normal space that is not limit point compact. Then there exists a closed infinite subset A X [Thm 17.6]. Let f : X R be the extension [Thm 35.1] of any surjection A Z + . Then f is unbounded. (iii) (i): Any limit point compact metrizable space is compact [Thm 28.2]; any metric on X is continuous [Ex 20.3], hence bounded [Thm 26.5]. Ex. 35.4. Let Z be a topological space and Y Z a subspace. Y is a retract of Z if the identity map on Y extends continuously to Z , i.e. if there exists a continuous map r : Z Y such that Y Y Z r commutes. (a) . Y = { z Z | r ( z ) = z } is closed if Z is Hausdorff [Ex. 31.5]. (b) . Any retract of R 2 is connected [Thm 23.5] but A is not connected. (c) . The continuous map r ( x ) = x/ | x | is a retraction of the punctured plane R 2 - { 0 } onto the circle S 1 R 2 - { 0 } . Ex. 35.5. A space Y has the UEP if the diagram (1) A f Y X f has a solution for any closed subspace A of a normal space X . (a) . Another way of formulating the Tietze extension theorem [Thm 35.1] is: [0 , 1], [0 , 1), and (0 , 1) R have the UEP. By the universal property of product spaces [Thm 19.6], map( A, X α ) = map( A, X α ), any product of spaces with the UEP has the UEP. (b) . Any retract Y of a UEP space Z is a UEP space for in the situation A f Y Z r X f the continuous map r f : X Y extends f : A Y .
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