GENERALIZED FRÉCHET-URYSOHN SPACES

Spaces which are generated by discrete sets

UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn countably compact topological space.

On embeddings of the countable fan space

A NATURAL GENERALIZATION OF CHABER'S THEOREM

A first countable compact space that is not an N∗ image

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

We prove that if K is a compact space and the space P(K K) of regular probability measures on K K has countable tightness in its weak topology, then L1( ) is separable for every 2 P(K). It has been known that such a result is a consequence of Martin's axiom MA(!1). Our theorem has several consequences; in particular, it general- izes a theorem due to Bourgain and Todor£evi¢ on measures on Rosenthal compacta. 1. Introduction. The tightness of a topological space X, denoted by (X), is the least cardinal number such that for every A X and x2 A there is a set A0 A withjA0j (X) and such that x2 A0. Throughout, K stands for a compact Hausdor topological space. By

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.

Extensions of sufficient conditions for the product of two countably compact spaces to be countably compact, plus a relevant example.

A pseudocompact group which is not strongly pseudocompact

It is proved that every infinite dimensional separable Banach space having the separable extension property is isomorphic to c0. It is also proved that every Banach space with a separable dual is “close” to a space of continuous functions on a countable compact space.

A topological property \(\mathscr {P}\) is reflected in continuous images of weight at most \(\omega _1\) if a space X has \(\mathscr {P}\) whenever every continuous image of X of weight at most \(\omega _1\) has \(\mathscr {P}\). When X is a generalized ordered space, we consider a number of topological properties including feeble Lindelofness and \(\kappa \)-monolithicity. In the final section we study small images of pseudocompact and countably compact spaces; we give a condition on continuous images of a pseudocompact space in order that it be compact and show that it is consistently true that a countably compact space of countable projective tightness is countably tight.

In this paper two ways of generalizing compactness are studied.We may consider various types of refinements of open covers, such as countable open refinements, locally finite open refinements, etc.In another direction, countably compact spaces may be characterized as having the property that any sequence has a cluster point.Spaces which require that certain sequences have cluster points, such as ^-spaces, mJ-spaces, and g-spaces, will be referred to as generalized countably compact spaces.

In this paper, we introduce closure operators [<TEX>${\cdot}$</TEX>]c and [<TEX>${\cdot}$</TEX>]a on a space and study some relations among [<TEX>${\cdot}$</TEX>]c, [<TEX>${\cdot}$</TEX>]a and countable tightness. We introduce the concepts of a strongly sequentially closed set and a strongly sequentially open set and show that a space X has countable tightness if and only if every strongly sequentially closed set is closed if and only if every strongly sequentially open set is open. Finally we find a generalization of the weak Fr<TEX>$\'{e}$</TEX>chet-Urysohn property which is equivalent to countable tightness.

ABSTRACT: An ideal on a set X is a nonempty collection of subsets closed under the operations of finite union and subset. The concepts of parabounded and countably bounded subsets are defined as well as regularity with respect to an ideal I(i.e., I‐regular). The bounded subsets of a topological space are characterized as the subsets which are both parabounded and countably bounded. It is shown that if (X, τ) is a Hausdorff space and I is an ideal on X such that I∩τ={Ø}, then (X, τ) is compact modulo I iff (X, τ) is H‐closed and I‐regular. Examples are given which place e‐Lindelöfness in the hierarchy of known properties and it is shown that para‐Lindelöf e‐Lindelöf spaces are Lindelöf. It is shown that in the category of Hausdorff spaces, perfect images of e‐paracompact spaces are e‐paracompact. Posed as an open question is the following: What can be said about the closed continuous image of an e‐paracompact space?