Hausdorffness is hereditary

This article gives the statement, and possibly proof, of a topological space property (i.e., Hausdorff space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Statement

Any subspace of a Hausdorff space is Hausdorff in the subspace topology.

Definitions used

Hausdorff space

Further information: Hausdorff space

A topological space $X$ is Hausdorff if given distinct points $a,b\in X$ there exist disjoint open subsets $U,V$ containing $a,b$ respectively.

Subspace topology

Further information: subspace topology

If $A$ is a subset of $X$, we declare a subset $V$ of $A$ to be open in $A$ if $V=U\cap A$ for an open subset $U$ of $X$.

Proof

Given: A topological space $X$, a subset $A$ of $X$. Two distinct points $x_{1},x_{2}\in A$.

To prove: There exist disjoint open subsets $U_{1},U_{2}$ of $A$ such that $x_{1}\in U_{1},x_{2}\in U_{2}$.

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$x_{1},x_{2}$ are distinct points of $X$		$x_{1},x_{2}$ are distinct points of $A$ $A\subseteq X$
2	There exist disjoint open subsets $V_{1},V_{2}$ of $X$ such that $x_{1}\in V_{1},x_{2}\in V_{2}$.		$X$ is Hausdorff	Step (1)	Step-given direct
3	Define $U_{1}=V_{1}\cap A$ and $U_{2}=V_{2}\cap A$.
4	$U_{1},U_{2}$ are open subsets of $A$.	definition of subspace topology		Steps (2), (3)	By Step (2), $V_{1},V_{2}$ are open, so by the definition of subspace topology, $U_{1},U_{2}$ are open as per their definitions in Step (3).
5	$U_{1},U_{2}$ are disjoint.			Steps (2), (3)	follows directly from $V_{1},V_{2}$ being disjoint
6	$x_{1}\in U_{1},x_{2}\in U_{2}$		$x_{1},x_{2}\in A$	Steps (2), (3)	By Step (3), $U_{1}=V_{1}\cap A$. By Step (2), $x_{1}\in V_{1}$, and we are also given that $x_{1}\in A$, so $x_{1}\in V_{1}\cap A=U_{1}$. Similarly, $x_{2}\in U_{2}$.
7	$U_{1},U_{2}$ are the desired open subsets of $A$.			Steps (4)-(6)	Step-combination direct, it's what we want to prove.

This proof uses a tabular format for presentation. Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

References

Textbook references

• Topology (2nd edition) by James R. Munkres, ^{More info}, Page 100, Theorem 17.11, Page 101, Exercise 12 and Page 196 (Theorem 31.2 (a))