We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...

The problem whether every topological space $X$ has a compactification $Y$ such that every continuous mapping $f$ from $X$ into a compact space $Z$ has a continuous extension from $Y$ into $Z$ is answered in the negative. For some spaces $X$ such compactifications exist.

Given a discrete group $G$, we consider the set $ℒ\left(G\right)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ\left(G\right)$ into the Cantor cube ${\{0,1\}}^G$. We show that the cellularity $c\left(ℒ\left(G\right)\right)\le {\aleph }_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c\left(ℒ\right(G\left)\right)=\tau$.

A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d(f^n\left(x\right),f^n\left(y\right))>c$ (resp. $diam{f}^n\left(A\right)>c$). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum...

It is shown that the hyperspace $Cl{d}_H\left(X\right)$ (resp. $Bd{d}_H\left(X\right)$) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

Realcompact spaces can be characterized as limits of approximate inverse systems of Polish polyhedra.

The main result of the present paper is a classification theorem for finite-sheeted covering mappings over connected paracompact spaces. This theorem is a generalization of the classical classification theorem for covering mappings over a connected locally pathwise connected semi-locally 1-connected space in the finite-sheeted case. To achieve the result we use the classification theorem for overlay structures which was recently proved by S. Mardesic and V. Matijevic (Theorems 1 and 4 of [5]).

Clone properties are the properties expressible by the first order sentence of the clone language. The present paper is a contribution to the field of problems asking when distinct sentences of the language determine distinct topological properties. We fully clarify the relations among the rigidity, the fix-point property, the image-determining property and the coconnectedness.

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property $\left(a\right)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $\left(a\right)$) in $X$ are necessarily countable. There are, however, consistent examples of...

In some sense, a dual property to that of Valdivia compact is considered, namely the property to be embedded as a closed subspace into a complement of a $\Sigma$-subproduct of a Tikhonov cube. All locally compact spaces are co-Valdivia spaces (and only those among metrizable spaces or spaces having countable type). There are paracompact non-locally compact co-Valdivia spaces. A possibly new type of ultrafilters lying in between P-ultrafilters and weak P-ultrafilters is introduced. Under Martin axiom and...

We prove some closed mapping theorems on $k$-spaces with point-countable $k$-networks. One of them generalizes Lašnev’s theorem. We also construct an example of a Hausdorff space $Ur$ with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a $k$-space $X$ with a point-countable $k$-network admitting a closed surjection which is not compact-covering contains a closed copy of $Ur$.