Obtainable Sizes of Topologies on Finite Sets

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.Comment: Final version, to appear in Journal of Combinatorial Theory, Series

S and L Spaces

An S-space is any topological space which is hereditarily separable but not Lindelof. An L-space, on the other hand, is hereditarily Lindelof but not separable. For almost a century, determining the necessary and suffcient conditions for the existence of these two kinds of spaces has been a fruitful area of research at the boundary of topology and axiomatic set theory. For most of that time, the twoproblems were imagined to be dual; that is, it was believed that the same setsof conditions that required or precluded one type would suffice for the other aswell. This, however, is not the case. When Todorcevic proved in 1981 that itis consistent, under ZFC, for no S-spaces to exist, everyone expected a similarresult to follow for L-spaces as well. Justin Tatch Moore surprised everyonewhen, in 2005, he constructed an L-space in ZFC. This paper summarizes andcontextualizes that result, along with several others in the field

Generating some large filters of quasiorder lattices

In 2021, the author pointed out that there are some connections between the present topic and cryptography, and his parallel paper on Boolean lattices sheds more light on these connections. Earlier, his 2017 paper proved that for all but three values of a finite number $n$, the lattice of quasiorders (AKA preorders) of an $n$-element set has a four-element generating set. Here we prove that some large filters of the lattices of quasiorders of finite sets (and some infinite sets) have small generating sets, too; some of these filters occurring in the paper consist of the extensions of the partial orders of forests with few edges.Comment: 18 pages, 2 figures. Due to arXiv:2303.10790, now the main result is stronger than that in the earlier versio

Locally compact, ω1\omega_1-compact spaces

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Comment: 21 pages, submitted, comments are welcom

A new approach on distributed systems: orderings and representability

In the present paper we propose a new approach on `distributed systems': the processes are represented through total orders and the communications are characterized by means of biorders. The resulting distributed systems capture situations met in various fields (such as computer science, economics and decision theory). We investigate questions associated to the numerical representability of order structures, relating concepts of economics and computing to each other. The concept of `quasi-finite partial orders' is introduced as a finite family of chains with a communication between them. The representability of this kind of structure is studied, achieving a construction method for a finite (continuous) Richter-Peleg multi-utility representation