Extension of the mappings to the topological groups
by
Nazar Pyrch
Ukraine Academy of Printing

A subspace Y of a topological space X is called a G-retract of X, if any continuous mapping from Y to a Hausdorff topological group G admits a continuous extension on X.

Proposition 1 Let X be a topological space, r₁, r₂:X→ X be retractions such that r₁○r₂(X)=r₂○r₁(X). Then the subspace K=r₁(X)∪r₂(X) is G-retract of topological space X.

Proposition 2 Let Y be a subspace of a functionally Hausdorff space X. Let (F(X), h_X), (F(Y), h_Y) be the free topological groups on X and Y (in the sense of [1]). Then the following are equivalent:

1) Y is a G-retract in X;

2) there exists a homomorphism h: F(X)→ F(Y) such that h(h_X(y))=h_Y(y) for all y ∈ Y.

It follows from this proposition that any G-retract of functionally Hausdorff topological space X is closed in X.

Topological space X is called an absolute G-retract, if X is a G-retract of any topological space Y containing X as a closed subspace.

Proposition 3 Let X be a topological space A be a closed subspace of X such that topological spaces A and X/A are absolute G-retracts. Then the topological space X is an absolute G-retract.

[1] Fay T.H., Ordman E.T., SmithThomas B.V. The free topological groups over rationals // General Topology and its Applications 10 (1979), 33-47.

Date received: April 22, 2008