§ 6. 3 Urysohn introduction and tietze extension theorem

§ 6. 3 Urysohn introduction and tietze extension theorem

This section focuses on:

Understand the content of Urysohn's theorem (not required for proof );

Master the proof method of Theorem 6.3.2.

Theorem 6.3.1 [Urysohn theorem] sets X as a topological space and [a, B] as a closed interval. then X is a formal space. If and only for any two closed sets A and B in X, a continuous ing F: X → [, b] makes f (x) = A when X is a and f (x) = B when X is B.

Proof (omitted)

Theorem 6.3.2 if any connected subset in a space contains more than one vertex, it must be an indispensable set.

It is proved that C is a connected subset of space X. if C contains more than one vertex, select any vertex, X, Y, X, X, and Y. For two non-intersection closed sets {x} and {y} in space X }, the Urysohn theorem is applied to show that there is a continuous ing F: X → [0, 1] So that f (x) = 0, f (y) = 1. since C is a connected subset of X, f (x) is also connected. f (x) = [0, 1]. [0, 1] is an indispensable set, so C is also an indispensable set.