Arhangel'skiĭ sheaf amalgamations in topological groups
Volume 232 / 2016
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’skiĭ’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 ${\rm C_{p}}(X)$ of continuous real-valued functions on $X$ with the topology of pointwise convergence has Arhangel’skiĭ’s property $\alpha _1$ but is not countably tight. This follows from results of Arhangel’skiĭ–Pytkeev, Moore and Todorčević, and provides a new solution, with stronger properties than the earlier solution, of a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces.
