HYPERSPACE OF FINITE UNIONS OF CONVERGENT SEQUENCES

The symbol 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 S(X). Then we consider some cardinal invariants on S (X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on 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 S (X). Finally, we pose some questions about the hyperspace S (X).