A generalization of -metrizable spaces

2016

We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\varkappa$-metrizable spaces. We prove a generalization of a Chigogidze result that the \v{C}ech-Stone compactification of a pseudocompact countably $\varkappa$-metrizable space is $\varkappa$-metrizable.

Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1973

Some remarks on pseudocompact spaees A topological space E is pseuclocornpact if it satisfies one of the follorving equivalent conditions: (Pf) Every continuous real function of E is bounded, (P2) If /: Z-+ [0,1] is continuous, then /(,8) is closed ([10]), (PB) Evory countable cozero-set cover of E has a finite subcover ([6]), (P4) Every locally finite collection of cozero-sets of Z is finite ([6]), (P5) Every continuousl) pseudometric of ,E is precompact ([5]). Moreover, a uniformisable topological space E is pseudocompact if and only if (P6)'Every compatible uniformity of E is precompact ([B]). We generalize (P6)' to arbitrary topological spaces and present trvo conditions similar to (P5). Proposition 7 E or (L topolog,icctl space eclrc,iuulent: (P) fr ,is pseuiloco,nxpcr,ct, (P6) Eaery cont,i,n%ous uruifornt,ity ,f E (Pi) Eaery cont,inu,ou,s pseudometrio ,f E (P8) tuery continuoxLs pseudom,etric ,f E fr , tlte followi,tt g coyt d,,ition s &re 'is precom,pact.

Applied General Topology, 2013

The purpose of this paper is to introduce the notion of near metrizability for topological spaces, which is strictly weaker than the concept of metrizability. A number of characterizations of nearly metrizable spaces is achieved here as analogues of the corresponding ones for metrizable spaces. It is seen that near metrizability is a natural idea visa -vis near paracompactness, playing the similar role as played by paracompactness with regard to metrizability.

Recent Progress in General Topology III, 2013

International Journal of Mathematics and Mathematical Sciences, 2006

We apply the theory of the mutual compactificability to some spaces, mostly derived from the real line. For example, any noncompact locally connected metrizable generalized continuum, the Tichonov cube without its zero point I ℵ0 \{0}, as well as the Cantor discontinuum without its zero point D ℵ0 \{0} are of the same class of mutual compactificability as R.

Given a nonempty set X and a function f : X 6X, three pseudo-metrizable spaces are introduced. Some properties of these spaces and relations among them are studied and discussed.

Topology and its Applications, 2014

A topological space X is called hereditarily supercompact if each closed subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali, Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is hereditarily supercompact. A dyadic compact space is hereditarily supercompact if and only if it is metrizable. Under (MA+¬CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelöf. This implies that under (MA+¬CH) a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. The hereditary supercompactness is not productive: the product [0, 1] × αD of the closed interval and the one-point compactification αD of a discrete space D of cardinality |D| ≥ non(M) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof(M) = ω1 the space [0, 1] × αD contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.

Topology and its Applications, 1999

We prove some basic properties of p-bounded subsets (p ∈ ω *) in terms of z-ultrafilters and families of continuous functions. We analyze the relations between p-pseudocompactness with other pseudocompact like-properties as p-compactness and α-pseudocompactness where α is a cardinal number. We give an example of a sequentially compact ultrapseudocompact α-pseudocompact space which is not ultracompact, and we also give an example of an ultrapseudocompact totally countably compact α-pseudocompact space which is not q-compact for any q ∈ ω * , answering affirmatively to a question posed by S. García-Ferreira and Kocinac (1996). We show the distribution law cl γ (X×Y) (A × B) = cl γ X A × cl γ Y B, where γ Z denotes the Dieudonné completion of Z, for p-bounded subsets and we generalize the classical Glisckberg Theorem on pseudocompactness in the realm of p-boundedness. These results are applied to study the degree of pseudocompactness in the product of p-bounded subsets.

arXiv (Cornell University), 2017

Let us denote by Φ(λ, µ) the statement that B(λ) = D(λ) ω , i.e. the Baire space of weight λ, has a coloring with µ colors such that every homeomorphic copy of the Cantor set C in B(λ) picks up all the µ colors. We call a space X π-regular if it is Hausdorff and for every nonempty open set U in X there is a non-empty open set V such that V ⊂ U . We recall that a space X is called feebly compact if every locally finite collection of open sets in X is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let X be a crowded feebly compact π-regular space and µ be a fixed (finite or infinite) cardinal. If Φ(λ, µ) holds for all λ < c(X) then X is µ-resolvable, i.e. contains µ pairwise disjoint dense subsets. (Here c(X) is the smallest cardinal κ such that X does not contain κ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill [6], resp. Ortiz-Castillo and Tomita .

Topology and its Applications, 1999

Let X and Y be Hausdorff topological spaces. Let P be the family of all partial maps from X to Y: a partial map is a pair (B,f). where B E CL(X) (= the family of all nonempty closed subsets of X) and f is a continuous function from B to El'. Denote by 7~ the generalized compact-open topology on P. We show that if X is a hemicompact metrizable space and Y is a FrCchet space. then (P. TC) is completely metrizable and homeomorphic to a closed subspace of (CL(X), TF) x (C(X. Y). T~,cJ), where T,T is the Felt topology on CL(X) and 71'0 is the compact-open topology on C(X, Y).