On Hereditarily Normal Topological Groups

TL;DR

This paper studies hereditarily normal topological groups, proving that their compact subspaces are metrizable and exploring conditions under which countably compact subsets are metrizable, with implications under certain set-theoretic axioms.

Contribution

It establishes that compact subspaces of hereditarily normal topological groups are metrizable and links the existence of non-trivial convergent sequences to $G_\delta$-diagonals, advancing understanding of their structure.

Findings

Every compact subspace of a hereditarily normal topological group is metrizable.

A hereditarily normal topological group with a non-trivial convergent sequence has a $G_\delta$-diagonal.

Under the Proper Forcing Axiom, all countably compact subsets are metrizable.

Abstract

In this paper we investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a hereditarily normal topological group with a non-trivial convergent sequence has $G_\delta$-diagonal. This implies, in particular, that every countably compact subset of a hereditarily normal topological group with a non-trivial convergent sequence is metrizable. Another corollary is that under the Proper Forcing Axiom, every countably compact subset of a hereditarily normal topological group is metrizable.