We extend the Noble and Ulmer theorem and the Juhász and Hajnal theorems in set-theoretic topology. We show that a statement analogous to that in the former theorem is valid for a family of almost topological convergences, whereas statements analogous to those in the latter theorems hold for a pretopologically Hausdorff convergence.
Let be a topological property. A space is said to be star if whenever is an open cover of , there exists a subspace with property such that , where In this paper, we study the relationships of star properties for in pseudocompact spaces by giving some examples.
We provide a further estimate on the cardinality of a power homogeneous space. In particular we show the consistency of the formula for any regular power homogeneous ccc space.
Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra...
Every crowded space is -resolvable in the c.c.c. generic extension of the ground model. We investigate what we can say about -resolvability in c.c.c. generic extensions for . A topological space is monotonically -resolvable if there is a function such that for each . We show that given a space the following statements are equivalent: (1) is -resolvable in some c.c.c. generic extension; (2) is monotonically -resolvable; (3) is -resolvable in the Cohen-generic extension ....