# Countably z-compact spaces

## A. T. Al-Ani

Address: Department of Mathematics, College of Science and Information Technology, Irbid National University, Irbid, Jordan

E-mail: atairaqi@yahoo.com

Abstract: In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions are given.

AMSclassification: primary 54C60; secondary 54D30.

Keywords: z-compact space, z-Lindelof space, compact space, pseudocompact space, realcompact space.

DOI: 10.5817/AM2014-2-97