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.
Let C(X) denote the hyperspace of subcontinua of a continuum X. For p ∈ X, define the hyperspacesConditions under which it is compact, connected, arcwise connected and locally connected are studied. A characterization of hereditarily indecomposable continua is also given.
We continue the study of 1 2 -homogeneity of the hyperspace suspension of continua. We prove that if X is a decomposable continuum and its hyperspace suspension is 1 2 -homogeneous, then X must be continuum chainable. We also characterize 1 2 -homogeneity of the hyperspace suspension for several classes of continua, including: continua containing a free arc, atriodic and decomposable continua, and decomposable irreducible continua about a finite set.
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