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Every regular space is hereditarily regular.

Every regular space is hereditarily regular.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you explain how to show 4.20 in easiest possible way(in very detail)?

Theorem 4.20. Every regular space is hereditarily regular. Definition. Let P be a topological property (such as T1, Hausdorff, etc.). A topological space X is hereditarily P if and only if for each subspace Y of X, the space Y has property P when Y is given the relative topology from X. X is regular if and only if for every point x E X and closed set A C X not containing x, there are disjoint open sets U,V such that x E U and A C V. A T3-space is any space that is both T¡ and regular.
Transcribed Image Text:Theorem 4.20. Every regular space is hereditarily regular. Definition. Let P be a topological property (such as T1, Hausdorff, etc.). A topological space X is hereditarily P if and only if for each subspace Y of X, the space Y has property P when Y is given the relative topology from X. X is regular if and only if for every point x E X and closed set A C X not containing x, there are disjoint open sets U,V such that x E U and A C V. A T3-space is any space that is both T¡ and regular.
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