Hausdorff property is hereditary
Theorem 1.
A subspace of a Hausdorff space is Hausdorff.
Proof.
Let be a Hausdorff space , and let be a subspace of . Let where . Since is Hausdorff, there are disjoint neighborhoods of and of . Then is a neighborhood of in and is a neighborhood of in , and and are disjoint. Therefore, is Hausdorff. ∎
Title | Hausdorff property is hereditary |
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Canonical name | HausdorffPropertyIsHereditary |
Date of creation | 2013-03-22 15:22:27 |
Last modified on | 2013-03-22 15:22:27 |
Owner | georgiosl (7242) |
Last modified by | georgiosl (7242) |
Numerical id | 8 |
Author | georgiosl (7242) |
Entry type | Theorem |
Classification | msc 54D10 |