applications of Urysohn’s Lemma to locally compact Hausdorff spaces
denotes the set of continuous complex functions on ;
denotes the set of continuous and bounded complex functions on ;
denotes the set of continuous complex functions on which vanish at infinity;
denotes the set of continuous complex functions on with compact support
Note that we have , and that when we replace with (in general, when is compact), these four classes of functions coincide.
Now, while Urysohn’s Lemma does not directly apply to (since need not in general be normal), it does apply to , for being compact Hausdorff, is necessarily normal. One may therefore indirectly apply Urysohn’s Lemma to by way of to obtain various results asserting the existence of certain continuous functions on with prescribed properties. The following results and their proofs illustrate this technique and are frequently useful in analysis.
If with compact and open, then there exists an open subset of with compact closure such that .
Since is a compact subset of the Hausdorff space , it is closed, and because is open in , is as well. Therefore, by normality, there exists an open subset of such that (note that the closure of in coincides with that of in , since the former set is contained in and the latter set is equal to the former intersected with ). As is closed in , it is compact, and because is open in and , is open in . Thus possesses the desired properties. ∎
For each and each open subset of containing , there exists an open subset of with compact closure such that and .
Take in the preceding proposition. ∎
(Urysohn’s Lemma for LCH Spaces) If with compact and open, then there exists such that , , and .
By the first Proposition, there exists an open subset of with compact closure such that ; since and are disjoint closed subsets of the normal space , Urysohn’s Lemma furnishes such that , , and . Put . Then , , and . Moreover, vanishes outside because does, so ; since is compact, and consequently closed, the last inclusion gives and . ∎
(Tietze Extension Theorem for LCH Spaces) If is compact and is real, then there exists a real extending .
is the uniform closure of in .
We first show that is closed in . To this end, assume that is a uniformly convergent sequence of functions in with limit and let be given. Select such that , and select a compact subset of such that for . We then have, for all such ,
Thus vanishes at infinity; since the uniform limit of continuous functions is continuous, we obtain , whence is closed. It remains to establish the density of in . Given and , select a compact subset of such that for . By Theorem 1, there exists with range in satisfying . The function is continuous and supported inside , hence lies in ; moreover, if , then we have , while if , then
It follows that , hence that , completing the proof. ∎
|Title||applications of Urysohn’s Lemma to locally compact Hausdorff spaces|
|Date of creation||2013-03-22 18:33:31|
|Last modified on||2013-03-22 18:33:31|
|Last modified by||azdbacks4234 (14155)|