On sequentially closed subsets of the real line in |
| |
| Authors: | Kyriakos Keremedis |
| |
| Institution: | Department of Mathematics, University of the Aegean, Karlovassi, Greece |
| |
| Abstract: | We show: - (i)
 iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. - (ii) Every infinite subset X of
 has a countably infinite subset iff every infinite sequentially closed subset of  includes an infinite closed subset. - (iii) The statement “
 is sequential” is equivalent to each one of the following propositions: - (a) Every sequentially closed subset A of
 includes a countable cofinal subset C, - (b) for every sequentially closed subset A of
 ,  is a meager subset of  , - (c) for every sequentially closed subset A of
 ,  , - (d) every sequentially closed subset of
 is separable, - (e) every sequentially closed subset of
 is Cantor complete, - (f) every complete subspace of
 is Cantor complete. |
| |
| Keywords: | |
|
|
