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)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about Hausdorff space |Get facts that use property satisfaction of Hausdorff space | Get facts that use property satisfaction of Hausdorff space|Get more facts about product-closed property of topological spaces
Contents
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 is Hausdorff if given distinct points , there exist disjoint open sets containing and .
Product topology
Further information: Product topology
Suppose is an indexing set, and a family of topological spaces, . Then if is the Cartesian product of the s, the product topology on is a topology with subbasis given by all the open cylinders: all sets of the form such that for all but one , , and for the one exceptional , is an open subset of .
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)