point and a compact set in a Hausdorff space have disjoint open neighborhoods.
First we use the fact that is a Hausdorff space. Thus, for all there exist disjoint open sets and such that and . Then is an open cover for . Using this characterization of compactness (http://planetmath.org/YIsCompactIfAndOnlyIfEveryOpenCoverOfYHasAFiniteSubcover), it follows that there exist a finite set such that is a finite open cover for . Let us define
Next we show that these sets satisfy the given conditions for and . First, it is clear that and are open. We also have that and . To see that and are disjoint, suppose . Then for some . Since and are disjoint, can not be in , and consequently can not be in . ∎
- 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 2 I.M. Singer, J.A.Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967.
|Title||point and a compact set in a Hausdorff space have disjoint open neighborhoods.|
|Date of creation||2013-03-22 13:34:27|
|Last modified on||2013-03-22 13:34:27|
|Last modified by||drini (3)|