Hausdorffness is product-closed

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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_is, 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)