On Compact-covering Images of Locally Separable Metric Spaces

Abstract

Abstract. In this paper, we give the internal characterizations of compact-covering s-(resp., ˇ-)images of locally separable metric spaces. As applications of these results, weobtain characterizations of compact-covering quotient s-(resp., ˇ-)images of locally sepa-rable metric spaces. 1. IntroductionFinding the internal characterizations of certain images of metric spaces is aconsiderable interest in general topology. In the past, many nice results have beenobtained [6], [11], [12], [17], [18]. Recently, many topologists were engaged in re-search of internal characterizations of images of locally separable metric spaces,and some noteworthy results were shown. In [12], S. Lin, C. Liu, and M. Dai gavea characterization of quotient s-images of locally separable metric spaces. Afterthat, S. Lin, and P. Yan characterized sequence-covering s-images of locally sepa-rable metric spaces in [13]; Y. Ikeda, C. Liu and Y. Tanaka characterized quotientcompact images of locally separable metric spaces in [7]; and Y. Ge characterizedpseudo-sequence-covering compact images of locally separable metric spaces in [5].In a personal communication, the rst author of [12] and [13] informs that charac-terizations on compact-covering s-images and compact-covering ˇ-images still haveno answer. Thus, it is natural to rise the following question.Question 1.1. How are compact-covering s-(resp., ˇ-)images of locally separablemetric spaces characterized?In this paper, we give the internal characterizations of compact-covering s-(resp., ˇ-)images of locally separable metric spaces. As applications of these results,we obtain a characterization of compact-covering quotient s-(resp., ˇ-)images oflocally separable metric spaces.Throughout this paper, all spaces are assumed to be regular and T