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On preLindelöf metric spaces and the Axiom of Choice

On regular and homological closure operators

Observing that weak heredity of regular closure operators in Top and of homological closure operators in homological categories identifies torsion theories, we study these closure operators in parallel, showing that regular closure operators play the same role in topology as homological closure operators do algebraically.

On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even R may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of R, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that th...

The Ultrafilter Closure in ZF

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 by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem ...

What is a first countable space?

The definition of first countable space is standard and its meaning is very clear. But is that the case in the absence of the Axiom of Choice? The answer is negative because there are at least three choice-free versions of first countability. And, most likely, the usual definition does not correspond to what we want to be a first countable space. The three definitions as well as other characterizations of first...

Axioms for sequential convergence

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 sequential spaces. In case of unique limit points J. Kisynski [Kis60] obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to deal with. Finall...

On first and second countable spaces and the axiom of choice

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: ; http://www.sciencedirect.com/science/article/B6V1K-4C60730-4/1/ad5735139b6af6e881f45e6178f9793c

Sequential topological conditions in R in the absence of the axiom of choice

It is known that - assuming the axiom of choice - for subsets A of R the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in R, (c) R is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each ; http://dx.doi.org/10.1002/malq.200310029

On coregular closure operators

Among closure operators in the sense of Dikranjan and Giuli [5] the regular ones have a relevant role and have been widely investigated. On the contrary, the coregular closure operators were introduced only recently in [3] and they need to be further investigated. In this paper we study co regular closure operators, in connection connectednesses and disconnectednesses, in the realm of topological spaces a...