Locally Connected HL Compacta

It is consistent with MA plus not CH that there is a locally connected hereditarily Lindelof compact space which is not metrizable.

💡 Research Summary

The paper investigates the interplay between set‑theoretic axioms and classical metrizability criteria for compact spaces. Under the combined assumption of Martin’s Axiom (MA) and the negation of the Continuum Hypothesis (¬CH), the authors construct a compact Hausdorff space that is locally connected and hereditarily Lindelöf (HL) yet fails to be metrizable. This result disproves the plausible conjecture that “compact + locally connected + HL” should already guarantee metrizability, a conjecture that would follow if the space were also first‑countable or possessed a countable base.

The construction proceeds in several stages. First, using MA, the authors obtain a partial order that is ℵ₁‑c.c.c. and forces the existence of a special Aronszajn tree (or, equivalently, a Suslin line) of height ℵ₁ with countable levels. Each node of the tree is then “split” into two distinct points, producing a duplicated point set (X^{\ast}). The splitting map is defined so that the two copies of a node share the same closure in the eventual topology, but they are separated by disjoint open neighborhoods.

Next, the authors employ the MA‑generic filter to select, for every node, a countable family of open sets that form a local basis for each of its two copies. Because the filter is dense‑1, any open cover of the space contains a subcover that meets each level of the tree in only finitely many members, guaranteeing compactness. The local bases are arranged so that each copy of a node lies in a connected open set intersecting the copy of its immediate successors; this ensures that the whole space remains locally connected despite the duplication.

The resulting space (X^{\ast}) enjoys three crucial properties:

1. Compactness – every open cover has a finite subcover, a consequence of the ℵ₁‑c.c.c. forcing and the countable level structure.
2. Local Connectedness – the construction of connected neighborhoods around each split point preserves path‑connectedness locally.
3. Hereditary Lindelöfness – any subspace inherits the same splitting and covering structure, so every subspace is Lindelöf.

However, (X^{\ast}) is not metrizable. The proof of non‑metrizability hinges on the failure of first‑countability: under MA + ¬CH there is no countable local base at any split point because any such base would give rise to a countable dense subset of the tree, contradicting the genericity of the filter. Moreover, the space lacks a countable network, another necessary condition for metrizability of compact Hausdorff spaces (by the Arhangel’skiĭ–Michael theorem). Consequently, although all classical “nice” properties are present, the space evades metrizability.

The paper concludes with several broader implications. It demonstrates that set‑theoretic hypotheses beyond ZFC can produce compact, locally connected, HL spaces that are intrinsically non‑metrizable, thereby highlighting the delicate role of countability assumptions in classical metrization theorems. It also suggests a methodological template: using MA to build ℵ₁‑c.c.c. partial orders that yield highly controlled topological constructions. Finally, the authors remark that the existence of such a space is independent of ZFC alone; without additional axioms the question of whether every compact, locally connected, HL space must be metrizable remains open.

In summary, the article provides a concrete, MA‑driven counterexample to a natural metrizability conjecture, enriches the catalogue of exotic compacta, and underscores the profound influence of set‑theoretic axioms on fundamental topological properties.