Chains of simple closed curves and a dogbone space
Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke
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 (resp. ). 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...
Characterization of ultra separation axioms via -kernel.
Characterizations of the countable infinite product of rationals and some related problems
Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂
It is shown that the hyperspace (resp. ) 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.
Characterizing realcompact spaces as limits of approximate polyhedral systems
Realcompact spaces can be characterized as limits of approximate inverse systems of Polish polyhedra.
Classes of commutative rings characterized by going-up and going-down behavior
Classification of -cubes in the dimension
Classifying finite-sheeted covering mappings of paracompact spaces.
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 of topological spaces
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.
Closed copies of the rationals
Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property
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 . It follows that it is consistent that closed discrete subsets of a separable space which are also relatively normal (relatively countably paracompact, relatively ) in are necessarily countable. There are, however, consistent examples of...
Closed embeddings into complements of -products
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 -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...
Closed mapping theorems on -spaces with point-countable -networks
We prove some closed mapping theorems on -spaces with point-countable -networks. One of them generalizes Lašnev’s theorem. We also construct an example of a Hausdorff space with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a -space with a point-countable -network admitting a closed surjection which is not compact-covering contains a closed copy of .
Closed retractions of Euclidean spaces
Closed subsets of absolutely star-Lindelöf spaces II
In this paper, we prove the following two statements: (1) There exists a discretely absolutely star-Lindelöf Tychonoff space having a regular-closed subspace which is not CCC-Lindelöf. (2) Every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff) absolutely star-Lindelöf space as a closed subspace.
Closure, interior, and union in finite topological spaces
Cofinal completeness of the Hausdorff metric topology
A net in a Hausdorff uniform space is called cofinally Cauchy if for each entourage, there exists a cofinal (rather than residual) set of indices whose corresponding terms are pairwise within the entourage. In a metric space equipped with the associated metric uniformity, if each cofinally Cauchy sequence has a cluster point, then so does each cofinally Cauchy net, and the space is called cofinally complete. Here we give necessary and sufficient conditions for the nonempty closed subsets of the...
Coherent and strong expansions of spaces coincide
In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion...
Coincidence of Vietoris and Wijsman Topologies: A New Proof
Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric d forces X to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover we prove that (X, d) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on CL(X) coincide....