We say that a cardinal function $\phi$ reflects an infinite cardinal $\kappa$, if given a topological space $X$ with $\phi \left(X\right)\ge \kappa$, there exists $Y\in {\left[X\right]}^{\le \kappa }$ with $\phi \left(Y\right)\ge \kappa$. We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences with...

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition,...

We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular...

Every crowded space $X$ is $\omega$-resolvable in the c.c.c. generic extension $V^{Fn\left(\right|X|,2)}$ of the ground model. We investigate what we can say about $\lambda$-resolvability in c.c.c. generic extensions for $\lambda >\omega$. A topological space is monotonically ${\omega }_1$-resolvable if there is a function $f:X\to {\omega }_1$ such that $$\{x\in X:f\left(x\right)\ge \alpha \}{\subset }^{dense}X$$ for each $\alpha <{\omega }_1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\omega }_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically ${\omega }_1$-resolvable; (3) $X$ is ${\omega }_1$-resolvable in the Cohen-generic extension $V^{Fn({\omega }_1,2)}$....

We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger than the countable chain condition. It is proved that Martin's Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same can be forced...

We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of $2^{\omega }$ through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem...

Let $X$ be a uniform space of uniform weight $\mu$. It is shown that if every open covering, of power at most $\mu$, is uniform, then $X$ is fine. Furthermore, an ${\omega }_{\mu }$-metric space is fine, provided that every finite open covering is uniform.

The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$-th move the first player picks a point $x_n\in X$ and the second responds with choosing an open $U_n\ni x_n$. The game stops after $\omega$ moves and the first player wins if $\cup \{U_n:n\in \omega \}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta$ and $\Omega$. In $\theta$ the moves are made exactly as in the point-open game, but the...

A dense-in-itself space $X$ is called $C_{\square }$-discrete if the space of real continuous functions on $X$ with its box topology, $C_{\square }\left(X\right)$, is a discrete space. A space $X$ is called almost-$\omega$-resolvable provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa$-resolvable and almost resolvable spaces. We prove that every almost-$\omega$-resolvable space is $C_{\square }$-discrete, and that these classes coincide in...

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${\left[\omega \right]}^{\omega }$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.