For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_{\alpha }$ such that $ind{Z}_{\alpha }=\alpha$, and no closed subset L of $Z_{\alpha }$ with ind L less than the predecessor of α is a partition in $Z_{\alpha }$. An α-dimensional Cantor Ind-manifold can be constructed similarly.

Following Preuss' general connectedness theory in topological categories, a connectedness concept for approach spaces is introduced, which unifies topological connectedness in the setting of topological spaces, and Cantor-connectedness in the setting of metric spaces.

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space $X$ is maximally resolvable provided that for a dense set $X_0\subset X$ and for each $x\in X_0$ the $\pi$-character of $X$ at $x$ is not greater than the dispersion character of $X$. On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.

We prove that every compact space $X$ is a Čech-Stone compactification of a normal subspace of cardinality at most $d{\left(X\right)}^{t\left(X\right)}$, and some facts about cardinal invariants of compact spaces.