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 $g:B^n\to X$ such that ϱ(x,y) < ε, $\widehat{\varrho }(f,g)<\epsilon$ and $y\notin g\left(B^n\right)$. 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 $L{C}^{n-1}$-space then local homologies satisfy $H_k(X,X-x)=0$ for k < n and Hn(X,X-x) ≠ 0.

An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F₀,F₁ ⊂ X with f(F₀) = Y = f(F₁), provided all fibers of f are infinite and C*-embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.