Hausdorffness is hereditary

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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This article gives the statement, and possibly proof, of a basic fact in topology.

Statement

Property-theoretic statement

The property of topological spaces of being Hausdorff, is hereditary.

Verbal 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

Proof outline

The proof has the following key steps:

• Start with two points in the subspace
• View them as points in the whole space
• Separate them by disjoint open sets in the whole space
• Intersect these open sets with the subspace, and use the definition of subspace topology to note that we get disjoint open sets in the subspace separating the points