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Hausdorffness is product-closed
From Topospaces
This article gives the statement and proof of a property of topological spaces satisfying a metaproperty 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 X is Hausdorff if given distinct points 
, there exist disjoint open sets U,V containing a and b.
Product topology
- Further information: Product topology
. Then if X is the Cartesian product of the Xis, the product topology on X is a topology with subbasis given by all the open cylinders: all sets of the form | ∏ | Ai |
| 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. MunkresMore info, Page 101 (Exercise 11) and Page 196 (Theorem 31.2 (a))
