Hausdorffness is product-closed

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

Property-theoretic statement

The property of topological spaces of being a Hausdorff space is a product-closed property of topological spaces.

Verbal statement

An arbitrary (finite or infinite) product of Hausdorff spaces, when endowed with the product topology, is also a Hausdorff space.

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 sets $U,V$ containing $a$ and $b$.

Product topology

Further information: Product topology

Suppose $I$ is an indexing set, and $X_i$ a family of topological spaces, $i\in I$. Then if $X$ is the Cartesian product of the $X_i$s, the product topology on $X$ is a topology with subbasis given by all the open cylinders: all sets of the form ${\prod }_i{A}_i$ such that for all but one $i$, $A_i=X_i$, and for the one exceptional $i$, $A_i$ is an open subset of $X_i$.

Proof

Proof outline

The proof has the following three steps:

• Write down both points in the product space as tuples
• Find a coordinate where they differ, and separate the projections on that coordinate, by disjoint open sets in that coordinate (this is where we use that each space is Hausdorff)
• Use the open cylinders corresponding to these disjoint open subsets, to separate the original two points

References

Textbook references

• Topology (2nd edition) by James R. Munkres, ^{More info}, Page 100, Theorem 17.11, Page 101, Exercise 11, and Page 196, Theorem 31.2 (a)