Some steps on short bridges: Non-metrizable surfaces and CW-complexes

TL;DR

This paper investigates the homotopy types of non-metrizable surfaces, revealing that the presence of uncountably many infinitesimal bridges prevents such surfaces from being homotopy equivalent to CW-complexes, blending algebraic and set-theoretic topology techniques.

Contribution

It establishes a criterion linking infinitesimal bridges in non-metrizable surfaces to their homotopy equivalence with CW-complexes, using elementary algebraic and set-theoretic topology methods.

Findings

Surfaces with uncountably many infinitesimal bridges are not homotopy equivalent to CW-complexes.

Some classical Prüfer surfaces are homotopy equivalent to CW-complexes, others are not.

The presence of infinitesimal bridges obstructs CW-complex homotopy equivalence.

Abstract

Among the classical variants of the Pr\"ufer surface, some are homotopy equivalent to a CW-complex (namely, a point or a wedge of a continuum of circles) and some are not. The obstruction comes from the existence of uncountably many `infinitesimal bridges' linking two metrizable open subsurfaces inside the surface. We show that any non-metrizable surface that possesses such a system of infinitesimal bridges cannot be homotopy equivalent to a CW-complex. More than for the result on its own, we were motivated by trying to blend elementary techniques of algebraic and set-theoretic topology.