We show that the property of sequential compactness for subspaces of R is countably productive in ZF. Also, in the language of weak choice principles, we give a list of characterizations of the topological statement ‘sequentially compact subspaces of R are compact’. Furthermore, we show that forms 152 (= every non-well-orderable set is the union of a pairwise disjoint well-orderable family of d...

The concept of topological gyrogroups is a generalization group. In this work, ones prove that gyrogroup G metrizable iff has an {\omega}{\omega}-base and Frechet-Urysohn. Moreover, in gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn having are all weakly three-space properties with H closed L-subgyrogroup

Definition 1 (compare [2,5,8]) A topological space is called a KC-space provided that each compact set is closed. A topological space is called a U S-space provided that each convergent sequence has a unique limit. Remark 1 Each Hausdorff space (= T 2-space) is a KC-space, each KC-space is a U S-space and each U S-space is a T 1-space (that is, singletons are closed); and no converse implicatio...

We study weakly compact operators from a C∗-algebra with values in a complete locally convex space. They constitute a natural non-commutative generalization of finitely additive vector measures with values in a locally convex space. Several results of Brooks, Sâıto and Wright are extended to this more general setting. Building on an approach due to Sâıto and Wright, we obtain our theorems on no...

There are many topological results in Zermelo-Fraenkel set theory including the axiom of choice (ZFC) that are not true in the absence of choice, i. e. in ZF. Even if we restrict our attention to many “familiar” topological results are not provable in ZF, although in most cases their validity follows from a weaker version of the axiom of choice, CC( ). Definition 0.1 The axiom of countable choi...