Let X be a topological spaceMathworldPlanetmath, and C(X) the ring of continuous functions on X. A subspaceMathworldPlanetmath AX is said to be C-embedded (in X) if every function in C(A) can be extended to a function in C(X). More precisely, for every real-valued continuous functionMathworldPlanetmathPlanetmath f:A, there is a real-valued continuous function g:X such that g(x)=f(x) for all xA.

If AX is C-embedded, fg (defined above) is an embeddingMathworldPlanetmathPlanetmath of C(A) into C(X) by axiom of choiceMathworldPlanetmath, and hence the nomenclature.

Similarly, one may define C*-embedding on subspaces of a topological space. Recall that for a topological space X, C*(X) is the ring of boundedPlanetmathPlanetmathPlanetmathPlanetmath continuous functions on X. A subspace AX is said to be C*-embedded (in X) if every fC*(A) can be extended to some gC*(X).

Remarks. Let A be a subspace of X.

  1. 1.

    If A is C-embedded in X, and AYX, then A is C-embedded in Y. This is also true for C*-embeddedness.

  2. 2.

    If A is C-embedded, then A is C*-embedded: for if f is a bounded continuous function on A, say -nfn, and g is its continuous extensionPlanetmathPlanetmath on X, then -n(gn) is a bounded continuous extension of f on X.

  3. 3.

    The converseMathworldPlanetmath, however, is not true in general. A necessary and sufficient condition that a C*-embedded set A is C-embedded is:

    if a zero setPlanetmathPlanetmath is disjoint from A, the it is completely separated from A.

    Since any pair of disjoint zero sets are completely separated, we have that if A is a C*-embedded zero set, then A is C-embedded.


  • 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title C-embedding
Canonical name Cembedding
Date of creation 2013-03-22 16:57:37
Last modified on 2013-03-22 16:57:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54C45
Synonym C-embedded
Synonym C embedded
Synonym C*-embedded
Synonym C* embedded
Defines C-embedded
Defines C*-embedded