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

A NATURAL GENERALIZATION OF CHABER'S THEOREM

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.

In this paper, we introduce some new properties of a topological space which are respectively generalizations of -Urysohn property. We show that countably AP property is a sufficient condition for a space being countable tightness, sequential, weakly first countable and symmetrizable, to be ACP, , first countable and semimetrizable, respectively. We also prove that countable compactness is a sufficient condition for a countably AP space to be countably . We then show that a countably compact space satisfying one of the properties mentioned here is sequentially compact. And we show that a countably compact and countably AP space is maximal countably compact if and only if it is . We finally obtain a sufficient condition for the ACP closure operator to be a Kuratowski topological closure operator and related results.

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

Spaces which are generated by discrete sets

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

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

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.

Extending the class of known Stone–Čech remainders for ψ-spaces

In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

The present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle (see Qiu, 2014), we prove a new version of Ekeland variational principle (briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results (no matter scalar-valued case, or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.