All $\mathbb Q$-gluons are homeomorphic

Taras Banakh, Yaryna Stelmakh

Published 2020-03-13Version 1

In this paper we discover a (super)connected countable Hausdorf space called a $\mathbb Q$-gluon and prove its topogical uniqueness. A topological spaces $X$ is called $\mathbb Q$-$gluon$ if it possesses a decreasing sequence $(X_n)_{n\in\omega}$ of nonempty closed subsets such that $X_0=X$, $\bigcap_{n\in\omega}X_n=\emptyset$, for every $n\in\omega$ the complement $X\setminus X_{n+1}$ is homeomorphic to $\mathbb Q$, and for any non-empty open set $U\subseteq X_n$ the closure $\overline U$ contains some set $X_m$. Our main results says that any two $\mathbb Q$-gluons are homeomorphic. We shall find examples of $\mathbb Q$-gluons among quotient spaces, orbit spaces of group actions, and projective spaces of some topological vector spaces over countable topological fields. In particular, we prove that the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<\omega}=\{(x_n)_{n\in\omega}\in\mathbb Q^\omega:|\{n\in\omega:x_n\ne0\}|<\omega\}$ is a $\mathbb Q$-gluon.