$Z$-sets in ANR’s

Abstract:

(1) Let A be a closed Z-set in an ANR X. Let $\mathcal {F}$ be an open cover of X. Then there is a homotopy inverse $f:X \to X - A$ to the inclusion $X - A \to X$ such that f and both homotopies are limited by $\mathcal {F}$. (2) If, in addition, X is a manifold modeled on a metrizable locally convex TVS, F, such that F is homeomorphic to ${F^\omega }$, then there is a homotopy $j:X \times I \to X$ limited by $\mathcal {F}$ such that the closure (in X) of $j(X \times \{ 1\} )$ is contained in $X - A$.