C-5 - Cell-Like Maps

Abstract

A compact subset C of a space X is a cell-like set in X if for each neighborhood U of C the inclusion U :C →U is homotopic to a constant. Within absolute neighborhood retracts (ANRs), cell-likeness is an embedding invariant, not merely a positional feature relative to the superspace, because if C is cell-like in X, and ℯ: C→Y is an embedding into an ANR Y, then ℯ(C) is cell-like in Y. In addition, each neighborhood W of ℯ(C) then contains a smaller neighborhood W′ of ℯ(C) such that the inclusion W′→W is null-homotopic. Thus, a compact metric space can be said to be cell-like, provided under some (hence, under any) embedding in an ANR Y, its image is cell-like in Y. A compact, connected subset C of ℝ2, the Euclidean plane, is cell-like if and only if (iff) ℝ2 \\ C is connected. Generally, a compact metric space C is cell-like iff (1) it is acyclic with respect to Čech homology and (2) it has the following 1-UV Property: for some embedding ℯ: C →Q, the Hilbert cube, each neighborhood U of ℯ(C) contains a smaller neighborhood V of ℯ(C) such that every map ∂B2→V extends to a map B2 →U.