Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Stronger Urysohn lemma?

  1. Aug 14, 2009 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hello all,

    I am reading an article and there is something I find odd. The setting is a Banach space E and we have two disjoint closed subsets A and B of E. There is no additional assumption on E, A or B. The author then says,

    "Let f:E-->[0,1] be a Urysohn's function such that f(x)=0 if and only if x is in A, and f(x)=1 on B."

    But never have I seen a version of Urysohn's lemma that guarantees that f(x)=0 if and only if x is in A.

    Does someone have an explanation? (I would ask my advisor but she had gone on vacation for 3 weeks)
     
  2. jcsd
  3. Aug 14, 2009 #2
    In the exercises (exercise 5 on pg. 213) of Munkres' topology he states and asks the reader to prove the following theorem which he refers to as the strong form of the Urysohn lemma:
    Let X be a normal space. There is a continuous function [itex]f : X \to [0,1][/itex] such that f(x)=0 for [itex]x \in A[/itex], and [itex]f(x) = 1[/itex] for [itex]x\in B[/itex], and [itex]0 < f(x) < 1[/itex] otherwise, if and only if A and B are disjoint closed [itex]G_\delta[/itex] sets in X.

    In a metrizable space every closed set is [itex]G_\delta[/itex] and metrizable spaces are normal so we obtain the corollary:
    Let X be a metrizable space. Then there exists a continuous function [itex]f : X \to [0,1][/itex] such that f(x)=0 for [itex]x \in A[/itex], and [itex]f(x) = 1[/itex] for [itex]x\in B[/itex], and [itex]0 < f(x) < 1[/itex] otherwise, if and only if A and B are disjoint closed sets in X.
     
  4. Aug 14, 2009 #3

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I see, thank you!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Stronger Urysohn lemma?
  1. Konig Lemma (Replies: 2)

  2. Ito's Lemma (Replies: 6)

  3. What is a lemma? (Replies: 4)

Loading...