Cardinal functions of the hyperspace of convergent sequences

Abstract

The symbol 𝓢c(X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢c(X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$

“…Our next result is proved for infinite, non-discrete, Fréchet-Urysohn spaces in [10, Corollary 3.3]. However, using the proof of [17,Theorem 4.5] one can show it for arbitrary spaces. Theorem 1.10.…”

“…We note that Corollary 2.18, Theorem 2.20, Theorem 2.6 and Corollary 3.10 generalize significantly [9,Example 2.8], [17,Example 4.6], [9, Corollary 2.5] and [9, Lemma 2.11], respectively.…”

“…Since X is a continuum, it is clear that S c (X) is dense in CL(X). Furthermore, S c (X) is connected, by [17,Theorem 4.2]. Therefore, w(S c (X)) is connected and dense in [0, 1].…”

“…In connection with both concepts, the hyperspace consisting of all nontrivial convergent sequences S c (X), of a metric space X without isolated points, was introduced in 2015 in [9]. Since then, there has been increasing interest in studying S c (X) and several papers presenting relevant properties of this hyperspace have been written: [10], [11], [12], [16], [17] and [18].…”

“…Also, [9, Example 2.8] showed a pathwise connected continuum X such that S c (X) is not pathwise connected. Further, the authors in [9, Question 2.9] asked if S c (X) is pathwise connected when X is a dendroid; in [17,Example 4.6] this question was answered in the negative. They also asked whether the path connectedness of S c (X) implies that of X ([9, Question 2.14]); this question was answered in the affirmative for infinite, non-discrete, Fréchet-Urysohn spaces in [10,Corollary 3.3].…”