Hausdorff property is hereditary

Theorem 1.

Let X be a Hausdorff space , and let Y be a subspace of X. Let y1,y2Y where y1y2. Since X is Hausdorff, there are disjoint neighborhoodsMathworldPlanetmathPlanetmath U1 of y1 and U2 of y2 X. Then U1Y is a neighborhood of y1 in Y and U2Y is a neighborhood of y2 in Y, and U1Y and U2Y are disjoint. Therefore, Y is Hausdorff. ∎

Title Hausdorff property is hereditary
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