On the disjoint (0,N)-cells property for homogeneous ANRs

A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell $B^{n}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:B^{n} → X$ such that ϱ(x,y) < ε, $\widehat{ϱ}(f,g) < ε$ and $y ∉ g(B^{n})$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $LC^{n-1}$-space then local homologies satisfy $H_{k}(X,X-x) = 0$ for k < n and H_{n}(X,X-x) ≠ 0.