On Sequentially Connected Spaces

Sequentially connected space was defined by A.Fedeli and A.Le Donne. In this paper, we obtain characterizations of sequentially connected space by using 7-connected sets, give an example to illustrate that there exists a connected space but not sequentially connected and give the necessary and sufficient condition that connected space is sequentially connected. Morerover, we define the sequentially connected subsets, obtain several properties of them, and investigate the relationships between sequentially connected subsets and s-connected subsets, so we obtain the following results: (l)If S is a sequentially connected subset of X, then S is s-connected in X; If S is s-connected and sequentially closed in X, then S is sequentially connected in X; however, s-connected subset can't be sequentially connected subset. (2)If S be a sequentially connected subset of a space X and S C T C cs(S), then T is sequentially connected. Finally, we show that sequentially connectedness is countably multiplicative, discuss the properties of sequentially connected components, and give other description of sequentially connected space.