For sequential decision processes with countable state spaces, we prove compactness of the set of strategic measures corresponding to nonrandomized policies. For the Borel state case, this set may not be compact [14, p. 170] in spite of compactness of the set of strategic measures corresponding to all policies [17,2]. We use the compactness result from this paper to show the existence of optima...

We define several notions of rank-width for countable graphs. We compare, for each of them the width of a countable graph with the least upper-bound of the widths of its finite induced subgraphs. A width has the compactness property if these two values are equal. Our notion of rank-width that uses quasi-trees (trees where paths may have the order type of rational numbers) has this property. So ...

We investigate whether in the setting of approach spaces there exist measures of relative compactness, (relative) sequential compactness and (relative) countable compactness in the same vein as Kuratowski’s measure of compactness. The answer is yes. Not only can we prove that such measures exist, but we can give usable formulas for them and we can prove that they behave nicely with respect to e...

We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called recursively (countably) compact, if every recursive (respectively, countable) finitely consistent theory has a model. We show that for log...

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

abstract: A base B of a space X is called an OIF base when every element of B is a subset of only a finite number of other elements of B. We will explore the fundamental properties of spaces having such bases. In particular, we will show that in T2 spaces, strong OIF bases are the same as uniform bases, and that in T3 spaces where all subspaces have OIF bases, compactness, countable compactness...