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.