Definition 1 (compare [2,5,8]) A topological space is called a KC-space provided that each compact set is closed. A topological space is called a U S-space provided that each convergent sequence has a unique limit. Remark 1 Each Hausdorff space (= T 2-space) is a KC-space, each KC-space is a U S-space and each U S-space is a T 1-space (that is, singletons are closed); and no converse implicatio...

We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T1-space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally com...

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partially ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain so...

O. Wyler [Notices Amer. Math. Soc. 15 (1968), 169. Abstract #653-306.] has given a Stone-Cech compactification for limit spaces. However, his is not necessarily an embedding. Here, it is shown that any Hausdorff limit space (X, t) can be embedded as a dense subspace of a compact, Hausdorff, limit space (Xi, ri) with the following property: any continuous function from (X, t) into a compact, Hau...

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the ex...

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the ...

We prove that any metrizable non-compact space has a weaker metrizable nowhere locally compact topology. As a consequence, any metrizable non-compact space has a weaker Hausdorff connected topology. The same is established for any Hausdorff space X with a σ -locally finite base whose weight w(X) is a successor cardinal.  2002 Elsevier Science B.V. All rights reserved. AMS classification: Prima...

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, usual notion expansiveness for a homeomorphism compact metric space being particular case when lattice is topology phase ordered by inclusion automorphism one induced homeomorphism, mapping open sets to sets. prove this context generalizations Mañé's Theorem Utz's about finite dimensionality an...

In this paper we study spaces in which each compact subset is a Gδ-set and compare them to H. W. Martin’s c-semi-stratifiable (CSS) spaces, i.e. spaces in which compact sets are Gδ-sets in a uniform way. We prove that a (countably) compact subset of a Hausdorff space X is metrizable and a Gδ-subset of X provided X has a δθ-base, or a point-countable, T1-point-separating open cover, or a quasi-G...

We classify all one-point order-compactifications of a noncompact locally compact order-Hausdorff ordered topological space X. We give a necessary and sufficient condition for a one-point order-compactification of X to be a Priestley space. We show that although among the one-point order-compactifications of X there may not be a least one, there always is a largest one, which coincides with the...