Closed injective systems and its fundamental limit spaces

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.

💡 Research Summary

The paper introduces the notion of a closed injective system (CIS) of topological spaces and develops a comprehensive theory of its associated limit constructions. A CIS consists of a directed family ({X_i}{i\in I}) together with continuous injective bonding maps (f{ij}:X_i\to X_j) for (i<j) whose images are closed subsets of the target spaces. This closed‑image condition distinguishes CIS from the ordinary direct system used in classical direct limits.

Two kinds of limit objects are defined. The first, called the “limit space,” is the usual topological direct limit (\varinjlim X_i) obtained by taking the disjoint union of the (X_i) and imposing the final topology with respect to the bonding maps. The second, the “fundamental limit space,” is a minimal topological space in which each canonical inclusion (\iota_i:X_i\to L) is not only continuous but a topological embedding and the images remain closed. The authors prove a central existence‑and‑uniqueness theorem: every CIS admits a fundamental limit space, and it is unique up to homeomorphism. The proof uses Zorn’s Lemma to construct a partially ordered set of admissible topologies on the underlying set (\bigcup_i X_i) and shows that a minimal element exists and satisfies the required embedding property.

Having established the objects, the paper proceeds to a categorical formulation. Objects of the category (\mathcal{C}) are CISs; morphisms, called cis‑morphisms, are families of continuous maps that respect the directed structure and the closed‑image condition. The construction that assigns to each CIS its fundamental limit space extends to a functor (L:\mathcal{C}\to\mathbf{Top}). The functor preserves identities and composition, thereby embedding the CIS‑world into the ordinary category of topological spaces.

The authors also explore a “counter‑functor” perspective, attempting to reverse the direction of the bonding maps and form an inverse limit. They demonstrate that, because the closed‑image condition is not stable under reversal, a general inverse‑limit functor does not exist for CISs. Concrete counter‑examples (e.g., expanding intervals whose inverses fail to be closed) illustrate this obstruction, highlighting that CISs are intrinsically forward‑directed structures.

A substantial part of the work is devoted to “perfect properties,” i.e., topological properties such as compactness, completeness, or connectedness that are preserved by the bonding maps. The paper proves that if every bonding map in a CIS is a perfect map (continuous, closed, and with compact fibers), then the fundamental limit space inherits the same property. Conversely, dropping the closed‑image requirement can lead to loss of these properties in the limit, as shown by explicit pathological systems.

Several illustrative examples are presented: (1) an increasing chain of closed intervals (