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… (More)

In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T0 spaces via the (nowadays called) Scott topology, we study those metric spaces… (More)

It is of general knowledge that those (ultra)filter convergence relations coming from a topology can be characterized by two natural axioms. However, the situation changes considerable when moving to… (More)

Under the axiom of choice, every first countable space is a FréchetUrysohn space. Although, in its absence even R may fail to be a sequential space. Our goal in this paper is to discuss under which… (More)

In this paper it is studied the role of the axiom of choice in some theorems in which the concepts of first and second countability are used. Results such as the following are established: (1) In ZF… (More)

Topological sequential spaces are the fixed points of a Galois correspondence between collections of open sets and sequential convergence structures. The same procedure can be followed replacing open… (More)

It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters… (More)