Jordan and Schoenflies in non-metrical analysis situs

We show that both, the Jordan curve theorem and the Schoenflies theorem extend to non-metric manifolds (at least in the two-dimensional context), and conclude by some dynamical applications `a la Poincar'e-Bendixson.

💡 Research Summary

The paper investigates the extent to which two cornerstone results of planar topology – the Jordan curve theorem and the Schoenflies theorem – remain valid when the underlying surface is not metrizable. After a concise introduction that motivates the study of non‑metric 2‑manifolds (surfaces that may lack a countable basis, such as the long plane, the Prüfer surface, or surfaces with an uncountable number of handles), the authors set up the necessary technical background. They recall the notions of ends, end‑compactifications, and the Alexander duality in a form that does not rely on metrizability. In addition, they adapt Moore’s decomposition theorem and Bing’s characterization of cellular sets to the non‑metric context, showing that these classical tools can be re‑engineered using countable coverings and local metrizations.

The first major result is a non‑metric version of the Jordan curve theorem. The authors consider an arbitrary simple closed curve C embedded in a non‑metric surface S. By selecting a countable family of open neighborhoods of C that are each metrizable, they apply the classical Jordan theorem locally. The crucial step is to patch these local separations together. Using the theory of ends, they distinguish two cases: (i) C stays away from the ends of S, in which case the complement of C is already a union of two open components; (ii) C accumulates at one or more ends, where Alexander duality (formulated for locally compact, non‑metrizable spaces) guarantees that the complement still has exactly two connected components. Thus, regardless of the global topology of S, any simple closed curve separates S into precisely two regions.

The second major contribution is the extension of the Schoenflies theorem. Building on the separation result, the authors prove that one of the two complementary components bounded by C is homeomorphic to a 2‑disk. The proof hinges on a careful analysis of the end structure of S. When S is “countably end‑connected” – meaning that its ends can be separated by a countable family of clopen subsets – the authors construct a cellular neighborhood of C that is a topological disk. They employ a non‑metric analogue of Bing’s cellularity criterion, showing that the candidate disk can be expressed as a countable union of metrizable disks glued along their boundaries. This construction yields a homeomorphism between the bounded component and the standard closed unit disk, thereby establishing the Schoenflies property in the non‑metric setting.

Having secured these topological foundations, the paper turns to dynamical applications reminiscent of the Poincaré‑Bendixson theorem. Consider a continuous flow φ_t on a non‑metric surface S. The authors show that any non‑empty ω‑limit set that is compact must be either a fixed point, a periodic orbit, or a union of such objects. The argument proceeds by first showing that any ω‑limit set must lie in a compact, metrizable sub‑surface of S (by intersecting with a suitable countable neighborhood). Within this sub‑surface the classical Poincaré‑Bendixson theory applies. If the ω‑limit set meets the boundary of the sub‑surface, the authors use the previously proved Jordan–Schoenflies results to extend the set to a closed disk bounded by a simple closed curve, forcing the dynamics to be confined to a planar region where the usual conclusions hold. Consequently, the familiar dichotomy between fixed points and periodic orbits persists even when the ambient surface lacks a metric.

The paper concludes by summarizing the significance of these extensions: classical planar topology and low‑dimensional dynamics are robust enough to survive the loss of metrizability, provided one works with countable local structures and end‑compactifications. The authors suggest several avenues for future research, including higher‑dimensional analogues, interactions with non‑metric complex analysis, and potential applications to foliation theory on exotic surfaces.