Complete regularity is hereditary: Difference between revisions

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This article gives the statement, and possibly proof, of a topological space property (i.e., completely regular space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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This article gives the statement, and possibly proof, of a basic fact in topology.

Statement

Any subset of a completely regular space is completely regular in the subspace topology.

Definitions used

Term	Definition used
completely regular space	A space $X$ is completely regular if it is a T1 space and given any point $x\in X$ and closed subset $C\subseteq X$ such that $x\notin C$ , there exists a continuous map $f:X\to [0,1]$ such that $f(x)=0$ and $f(a)=1$ for all $a\in C$ .
subspace topology	For a subset $A$ of the space $X$ , the subspace topology on $A$ is defined as follows: a subset of $A$ is open in $A$ iff it can be expressed as the intersection with $A$ of an open subset of $X$ . Also, a subset of $A$ is closed in $A$ iff it can be expressed as the intersection with $A$ of a closed subset of $X$ .

Proof

Proof outline

Pick a point and a closed subset of the subspace
Find a closed subset of the whole space, whose intersection with the subspace is the given subset
Find a continuous function separating the point, and the bigger closed subset, in the whole space
Restrict this continuous function to the subspace, and observe that this works

References

Textbook references

Topology (2nd edition) by James R. Munkres, ^{More info}, Page 211-212, Theorem 33.2, Chapter 4, Section 33

@@ Line 14: / Line 14: @@
 ! Term !! Definition used
 |-
-| [[completely regular space]] || A space <math>X</math> is completely regular if it is a [[T1 space]] and given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.
+| [[completely regular space]] || A space <math>X</math> is completely regular if it is a [[T1 space]] and given any point <math>x \in X</math> and closed subset <math>C \subseteq X</math> such that <math>x \notin C</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in C</math>.
 |-
 | [[subspace topology]] || For a subset <math>A</math> of the space <math>X</math>, the subspace topology on <math>A</math> is defined as follows: a subset of <math>A</math> is open in <math>A</math> iff it can be expressed as the intersection with <math>A</math> of an open subset of <math>X</math>.<br>Also, a subset of <math>A</math> is closed in <math>A</math> iff it can be expressed as the intersection with <math>A</math> of a closed subset of <math>X</math>.