Hyperspace of finite unions of convergent sequences

TL;DR

This paper explores the properties of the hyperspace of finite unions of convergent sequences in a Hausdorff space, analyzing topological invariants, diagonal properties, and generalized metric relations.

Contribution

It provides a characterization of convergence in the hyperspace and compares various cardinal invariants of the hyperspace with those of the base space, introducing new insights into their relationships.

Findings

Characterization of convergence in $\\mathcal{S}(X)$

Comparison of cardinal invariants between $X$ and $\mathcal{S}(X)$

Existence of spaces with specific diagonal properties

Abstract

The symbol $\mathcal{S}(X)$ denotes the hyperspace of finite unions of convergent sequences in a Hausdorff space $X$. This hyperspace is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in $\mathcal{S}(X)$. Then we consider some cardinal invariants on $\mathcal{S}(X)$, and compare the character, the pseudocharacter, the $sn$-character, the $so$-character, the network weight and $cs$-network weight of $\mathcal{S}(X)$ with the corresponding cardinal function of $X$. Moreover, we consider rank $k$-diagonal on $\mathcal{S}(X)$, and give a space $X$ with a rank 2-diagonal such that $S(X)$ does not have any $G_{\delta}$-diagonal. Further, we study the relations of some generalized metric properties of $X$ and its hyperspace $\mathcal{S}(X)$. Finally, we pose some questions about the hyperspace $\mathcal{S}(X)$.