Question

Problem 5. Let X be a topological space. Assume that, for every pair of disjoint closed subsets E and F of X, there exists a continuous function f: X \to [0, 1] such that $E \subseteq f^{-1}(\{0\})$ and $F \subseteq f^{-1}(\{1\})$. Prove that X is normal. (This is the converse of the Urysohn lemma.)

Name: Problem 5. Let X be a topological space. Assume that, for every pair of disjoint closed subsets E and F of X, there exists a continuous function f: X →[0, 1] such that E ⊆ f^-1({0}) and F ⊆ f^-1({1}). Prove that X is normal. (This is the converse of the Urysohn lemma.)
Uploaded: 2023-08-29T02:19:39-08:00
Duration: 3 min 5 s
Channel: Madhur L
Description: Problem 5. Let X be a topological space. Assume that, for every pair of disjoint closed subsets E and F of X, there exists a continuous function f: X →[0, 1] such that E ⊆ f^-1({0}) and F ⊆ f^-1({1}). Prove that X is normal. (This is the converse of the Urysohn lemma.)

          Problem 5. Let X be a topological space. Assume that, for every pair of disjoint closed subsets E and F of X, there exists a continuous function f: X \to [0, 1] such that $E \subseteq f^{-1}(\{0\})$ and $F \subseteq f^{-1}(\{1\})$. Prove that X is normal. (This is the converse of the Urysohn lemma.)

$Problem 5. Let X be a topological space. Assume that, for every pair of disjoint closed subsets E and F of X, there exists a continuous function f: X →[0, 1] such that E ⊆ f^-1({0}) and F ⊆ f^-1({1}). Prove that X is normal. (This is the converse of the Urysohn lemma.)$

Added by Donald H.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Madhur L

08/29/2023

Step 1

By the given condition, there exists a continuous function $f: X \to [0, 1]$ such that $E \subset f^{-1}(\{0\})$ and $F \subset f^{-1}(\{1\})$. Show more…

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Problem 5. Let $X$ be a topological space. Assume that, for every pair of disjoint closed subsets $E$ and $F$ of $X$, there exists a continuous function $f: X \to [0, 1]$ such that $E \subset f^{-1}(\{0\})$ and $F \subset f^{-1}(\{1\})$. Prove that $X$ is normal. (This is the converse of the Urysohn lemma.)