1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
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...