On proper and exterior sequentiality

In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.

💡 Research Summary

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This paper develops a sequential theory tailored to the category of topological spaces equipped with proper maps and extends it to the broader setting of exterior spaces and exterior maps. The authors begin by recalling the classical notion of sequential spaces—those whose topology is completely determined by convergent sequences—and note that while sequential spaces enjoy many desirable categorical properties (coreflectivity, completeness, cartesian closedness), the analogous theory for proper maps has been lacking.

A proper map is defined as a continuous function f : X → Y such that the pre‑image of every closed compact subset of Y is closed compact in X. The authors introduce the category P of spaces and proper maps, and they observe that the Alexandroff one‑point compactification yields a full and faithful functor (–)⁺ : P → Top^∞, which becomes an equivalence when restricted to locally compact Hausdorff spaces.

The central new concept is that of a sequentially proper map. A map f : X → Y is called sequentially proper if (i) it is sequentially continuous (it sends convergent sequences to convergent sequences with the same limits) and (ii) it preserves proper sequences—those sequences s : ℕ → X for which the pre‑image of any closed compact set is finite. The authors prove that a map is sequentially proper precisely when its extension to the Alexandroff compactifications, f⁺ : X⁺ → Y⁺, is sequentially continuous. They also compare this definition with the earlier notion of B‑proper maps introduced by R. Brown, showing that on S₂‑spaces (sequential Hausdorff spaces) the two notions coincide.

To capture a class of spaces where properness and sequentiality interact nicely, the paper introduces s‑compact subsets: a subset C ⊂ X is s‑compact if it is sequentially closed and every proper sequence eventually avoids C. An ω‑sequential space is then defined as a sequential space in which the family of s‑compact subsets coincides with the family of closed compact subsets. Equivalently, the filter of open sets whose complements are s‑compact forms a basis for the topology. In such spaces, proper maps are exactly the sequentially proper maps (Proposition 3.6). The authors demonstrate that many familiar spaces—CW‑complexes, metric spaces, and smooth manifolds—are ω‑sequential, thereby showing that the new class is robust and includes the most commonly used examples.

Recognizing that the proper category P lacks certain categorical limits and colimits, the authors turn to exterior spaces, a framework introduced in earlier work that augments a topological space with a “neighbourhood system at infinity” (the externology). An exterior map is a continuous map that is continuous at infinity. The category E of exterior spaces is complete, cocomplete, and contains P as a full subcategory. The paper extends the ω‑sequential notion to this setting, defining e‑sequential exterior spaces. An e‑sequential space is an exterior space whose underlying topological space is ω‑sequential and whose externology is compatible with the s‑compact structure. The resulting subcategory E_seq is shown to be coreflective in E, a property that fails in the proper category. Moreover, E_seq embeds fully into a topos of sheaves, mirroring the classical result of Johnstone that the category of sequential spaces sits inside a sheaf topos.

The paper concludes by emphasizing that the sequential proper theory solves the open problems posed by R. Brown concerning when the one‑point compactification of a space is sequential. It also provides a unified language for dealing with non‑compact phenomena, offering tools that are likely to be valuable in areas such as shape theory, coarse geometry, and the study of dynamical systems on non‑compact spaces. Future work may explore homological invariants of e‑sequential spaces, connections with pro‑spaces, and applications to the categorical semantics of computation where “behaviour at infinity” plays a role.