On weak-open compact images of metric spaces

Zhiyun Yin

doi:10.1155/ijmms/2006/87840

Abstract

We give some characterizations of weak-open compact images of metric spaces.

Highlights

Introduction and definitionsTo find internal characterizations of certain images of metric spaces is one of central problems in general topology
The purpose of this paper is to give some characterizations of weak-open compact images of metric spaces, which showed that a space is a weak-open compact image of a metric space if and only if it has a weak development consisting of point-finite cs-covers
Let ᏼ be a cover of a space X. (1) ᏼ is called a cs-cover for X, if every convergent sequence in X is eventually in some element of ᏼ. (2) ᏼ is called an sn-cover for X, if every element of ᏼ is a sequential neighborhood of some point in X, and for any x ∈ X, there exists a sequential neighborhood P of x in X such that P ∈ ᏼ

Summary

Introduction

Introduction and definitionsTo find internal characterizations of certain images of metric spaces is one of central problems in general topology. (1) f is called a weak-open mapping [12], if there exists a weak base Ꮾ = ∪{Ꮾy : y ∈ Y } for Y , and for each y ∈ Y , there exists xy ∈ f −1(y) satisfying the following condition: for each open neighborhood U of xy, By ⊂ f (U) for some By ∈ Ꮾy. (2) P is called a sequential neighborhood of x in X, if whenever a sequence {xn} in X converges to x, {xn} is eventually in P. (3) P is called sequential open in X, if P is a sequential neighborhood at each of its points.

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