We answer questions of Bennett, Lutzer, and Matveev by showing that any monotonically compact LOTS is metrizable, and any first-countable Lindelöf GO-space is monotoncically Lindelöf. We also show that any compact monotonically Lindelöf space is first-countable, and is metrizable if scattered, and that separable monotonically compact spaces are metrizable.

We contribute to the refined understanding of language-logic-algebra interplay in context first-order properties countable words. establish decidable algebraic characterizations one variable fragment FO as well boolean closure existential via a strengthening Simon’s theorem about piecewise testable languages. propose new extension which admits infinitary quantifiers reason inherent provide very...

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Φ|| for sentences Φ of Lω1,ω, which counts the number of sentences of L∞,ω that, in some forcing extension, become a canonical Scott sentence of a model of Φ. We show this card...

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper an...

We use topological consequences of {\sf PFA}, MA$_{\omega_1}$(S)[S]} and PFA(S)[S]} proved by other authors to show that normal first countable linearly H-closed spaces with various additional properties are compact in these models.

In this paper, we study the conditions under which one can obtain a first countable and