Closed injective systems and its fundamental limit spaces

TL;DR

This paper introduces the concepts of limit space and fundamental limit space for closed injective systems of topological spaces, establishing their existence, uniqueness, and categorical properties, with applications to specific systems.

Contribution

It defines and explores the fundamental limit space for closed injective systems, including categorical frameworks and properties, advancing the theoretical understanding of these structures.

Findings

Existence and uniqueness of limit spaces established.

Categorical framework for closed injective systems developed.

Characterization results for specific fundamental limit spaces obtained.

Abstract

In this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. We present the main results on existence and uniqueness of limit spaces and several concrete examples. In the main section of the text, we show that the closed injective system can be considered as objects of a category whose morphisms are the so-called cis-morphisms. Moreover, the transition to fundamental limit space can be considered a functor from this category into category of topological spaces. Later, we show results about properties on functors and counter-functors for inductive closed injective system and fundamental limit spaces. We finish with the presentation of some results of characterization of fundamental limite space for some special systems and the study of so-called perfect properties.