One-step Closure, Ideal Convergence and Monotone Determined Space

TL;DR

This paper explores the properties of monotone determined spaces, focusing on one-step closure and ideal convergence, and characterizes c-spaces and locally hypercompact spaces within this framework.

Contribution

It introduces new characterizations of c-spaces and locally hypercompact spaces, and establishes conditions under which various convergence notions are topological.

Findings

Every c-space has one-step closure.

Locally hypercompact spaces have weak one-step closure.

IS-convergence and IGS-convergence are topological iff the space is a c-space or locally hypercompact.

Abstract

Monotone determined spaces are natural topological extensions of dcpo. Its main purpose is to build an extended framework for domain theory. In this paper, we study the one-step closure and ideal convergence on monotone determined space. Then we also introduce the equivalent characterizations of c-spaces and locally hypercompact space. The main results are:1.Every c-space has one-step closure and every locally hypercompact space has weak one-step closure;2.A monotone determined space has one-step closure if and only if it is d-meet continuous and has weak one-step closure. 3.IS-convergence(resp. IGS-convergence) is topological iff X is a c-space (resp. locally hypercompact space); 4.If X is a d-meet continuous space, then the following three conditions are equivalent to each other: (i) X is c-space; (ii) The net (xj ) ISL-converges to x iff (xj ) I-converges to x with respect to Lawson topology; (iii) The net (xj ) IGSL-converges to x iff (xj ) I-converges to x with respect to Lawson topology.