Let be a topological space, and the ring of continuous functions on . A subspace is said to be -embedded (in ) if every function in can be extended to a function in . More precisely, for every real-valued continuous function , there is a real-valued continuous function such that for all .
Similarly, one may define -embedding on subspaces of a topological space. Recall that for a topological space , is the ring of bounded continuous functions on . A subspace is said to be -embedded (in ) if every can be extended to some .
Remarks. Let be a subspace of .
If is -embedded in , and , then is -embedded in . This is also true for -embeddedness.
Since any pair of disjoint zero sets are completely separated, we have that if is a -embedded zero set, then is -embedded.
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
|Date of creation||2013-03-22 16:57:37|
|Last modified on||2013-03-22 16:57:37|
|Last modified by||CWoo (3771)|