The hyperspace of convergent sequences
In this paper, the authors introduce the notion of the hyperspace of nontrivial convergent sequences, and study its connectedness and Baire property. An infinite countable subset \(S\) of a space \(X\) is called a nontrivial convergent sequence if there exists \(x\in S\) such that \(S \setminus \{x\}\) is a discrete subspace and \(S \setminus U\) is finite for every neighborhood \(U\) of \(x\). The hyperspace \(\mathcal{S}_c(X)\) of nontrivial convergent sequences of a space \(X\) is the space consisting of all nontrivial convergent sequences in \(X\) with the Vietoris topology. The authors first prove that \(\mathcal{S}_c([0,1])\), \(\mathcal{S}_c(\mathbb{I})\) and \(\mathcal{S}_c([0,\omega_1])\) are not homeomorphic to each other, where \([0,1]\) is the closed interval in the space \(\mathbb{R}\) of real numbers, \(\mathbb{I}\) is the space of irrational numbers, and \([0,\omega_1)\) is the space of ordinals less than the first uncountable ordinal \(\omega_1\) with the order topology. Concerning the connectedness of \(\mathcal{S}_c(X)\), the following theorems are proved: \(\mathcal{S}_c([0,1])\) and \(\mathcal{S}_c(\mathbb{R})\) are path-wise connected. There exits a path-wise connected continuum \(X\) such that \(\mathcal{S}_c(X)\) is not path-wise connected. If \(X\) is path-wise connected, then \(\mathcal{S}_c(X)\) is connected. If \(X\) is a first countable space such that \(\mathcal{S}_c(X)\) is connected, then \(X\) is connected. Concerning the Baire property of \(\mathcal{S}_c(X)\), the authors prove that \(\mathcal{S}_c(X)\) does not satisfy the Baire property for any space \(X\), and \(\mathcal{S}_c(X)\) is a set of first category for every second countable \(X\). Several open questions on the space \(\mathcal{S}_c(X)\) are posed.
scientific article; zbMATH DE number 6522031