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

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))