product topology preserves the Hausdorff property
Theorem Suppose is a collection
of
Hausdorff spaces. Then the
generalized Cartesian product
equipped with the product topology is a Hausdorff space.
Proof. Let , and
let be distinct points in . Then there is an index
such that and are distinct points in
the Hausdorff space . It follows that there are open sets
and in such that , ,
and .
Let be the projection operator defined
here (http://planetmath.org/GeneralizedCartesianProduct). By the definition of
the product topology, is continuous, so
and are open sets in . Also,
since the
preimage
commutes with set operations
(http://planetmath.org/InverseImageCommutesWithSetOperations),
we have that
Finally, since , i.e., ,
it follows
that . Similarly, .
We have shown that and are open disjoint neighborhoods of
respectively . In other words, is a Hausdorff space.
Title | product topology preserves the Hausdorff property |
---|---|
Canonical name | ProductTopologyPreservesTheHausdorffProperty |
Date of creation | 2013-03-22 13:39:40 |
Last modified on | 2013-03-22 13:39:40 |
Owner | archibal (4430) |
Last modified by | archibal (4430) |
Numerical id | 7 |
Author | archibal (4430) |
Entry type | Theorem |
Classification | msc 54B10 |
Classification | msc 54D10 |