On selectively highly divergent spaces

TL;DR

This paper introduces the concept of selectively highly divergent (SHD) spaces, explores their properties, provides examples, and constructs new spaces with desired topological features related to SHD spaces.

Contribution

It defines SHD spaces, investigates their properties, and constructs examples and transformations that preserve or induce the SHD property in various topological contexts.

Findings

SHD spaces can have non-trivial convergent sequences and dense sets with no convergent sequences.

The $G_\delta$ modification of certain regular and homogeneous spaces is SHD and homogeneous.

A new extremally disconnected space $sX$ can be constructed from a Hausdorff space $X$ with preserved cardinal functions.

Abstract

We say that a topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open sets $\{U_n\mid n\in\omega \}$ of $X$, we can find $x_n\in U_n$ such that the sequence $(x_n)$ has no convergent subsequences. We investigate the basic topological properties of SHD spaces and we will exhibit that this class of spaces is full of variety. We present an example of a SHD space wich has a non trivial convergent sequence and with a dense set with no convergent sequences. Also, we prove that if $X$ is a regular space such that for all $x\in X$ holds $\psi(x,X)>\omega$, then $X_\delta$ (the $G_\delta$ modification of $X$) is a SHD space and, moreover, if $X$ homogeneous, then $X_\delta$ is also homogeneous. Finally, given $X$ a Hausdorff space without isolated points, we construct a new space denoted by $sX$ such that $sX$ is extremally disconnected, zero-dimensional Hausdorff space, SHD with $|X|=|sX|$, $\pi w(X)=\pi w(sX)$ and $c(X)=c(sX)$ where $\pi w$ and $c$ are the cardinal functions $\pi$-weight and celullarity respectively.