Arhangel'ski\u{\i} sheaf amalgamations in topological groups

TL;DR

This paper investigates amalgamation properties of convergent sequences in topological groups and vector spaces, establishing equivalences between certain properties and providing new solutions to longstanding problems in topology.

Contribution

It proves the equivalence of Nyikos's property α_{1.5} and Arhangel'ski's property α_1 in topological groups, and constructs a space with specific properties addressing a 1968 problem.

Findings

Nyikos's property α_{1.5} is equivalent to α_1 in topological groups

Existence of a space where C_p(X) has α_1 but is not countably tight

Provides a new solution to a problem on topological vector spaces from 1968

Abstract

We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos's property $\alpha_{1.5}$ is equivalent to Arhangel'ski\u{\i}'s formally stronger property $\alpha_1$. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space $X$ such that the space $C_p(X)$ of continuous real-valued functions on $X$, with the topology of pointwise convergence, has Arhangel'ski\u{\i}'s property $\alpha_1$ but is not countably tight. This result follows from results of Arhangel'ski\u{\i}--Pytkeev, Moore and Todor\v{c}evi\'c, and provides a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. The Averbukh--Smolyanov problem was first solved by Plichko (2009), using Banach spaces with weaker locally convex topologies.