On the disjoint (0,N)-cells property for homogeneous ANR's

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 ${\displaystyle B^n}$ into X and for each ε > 0 there exist a point y ∈ X and a map ${\displaystyle g:B^n\to X}$ such that ϱ(x,y) < ε, ${\displaystyle w\hat{\varrho }(f,g)<\epsilon }$ and ${\displaystyle y\notin g\left(B^n\right)}$. 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 ${\displaystyle L{C}^{n-1}}$-space then local homologies satisfy ${\displaystyle H_k(X,X-x)=0}$ for k < n and H_{n}(X,X-x) ≠ 0.