Hausdorffness is product-closed

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.

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
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 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

Textbook references

 * , Page 100, Theorem 17.11, Page 101, Exercise 11, and Page 196, Theorem 31.2 (a)