We prove resolvability and maximal resolvability of topological spaces having countable tightness with some additional properties. For this purpose, we introduce some new versions of countable tightness. We also construct a couple of examples of irresolvable spaces.

We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : $2^{\kappa }$ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered...

We prove that if there is an open mapping from a subspace of $C_p\left(X\right)$ onto $C_p\left(Y\right)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^{𝔱}>𝔠$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential.

Pointfree formulas for three kinds of separating points for closed sets by maps are given. These formulas allow controlling the amount of factors of the target product space so that it does not exceed the weight of the embeddable space. In literature, the question of how many factors of the target product are needed for the embedding has only been considered for specific spaces. Our approach is algebraic in character and can thus be viewed as a contribution to Kuratowski's topological calculus.

We show that the small cardinal number $i=min\left\{\right|𝒜|:𝒜$ is a maximal independent family} has the following topological characterization: $i=min\{\kappa \le c:{\{0,1\}}^{\kappa }$ has a dense irresolvable countable subspace}, where ${\{0,1\}}^{\kappa }$ denotes the Cantor cube of weight $\kappa$. As a consequence of this result, we have that the Cantor cube of weight $c$ has a dense countable submaximal subspace, if we assume (ZFC plus $i=c$), or if we work in the Bell-Kunen model, where $i={\aleph }_1$ and $c={\aleph }_{{\omega }_1}$.

Cardinal functions for topological spaces in which a subset is selected in a certain way are defined and studied. Most of the main cardinal inequalities are generalized for such spaces.

L'operazione «anti( )» di Paul Bankston fu introdotta in contesto della famiglia di tutti gli spazii topologici. Però, per molte ricerche ci conviene lavorare esclusivamente in una classe costretta di spazii di cui la struttura e ricca abbastanza di facilitare il ragionamento. In quest'articolo descriviamo come trasferire anti ( ), e concetti allacciati, dentro una tale classe costretta; con riferimento speciale all'esistenza di «pre-antis».

In this paper, two cardinal inequalities for functionally Hausdorff spaces are established. A bound on the cardinality of the $\tau \theta$-closed hull of a subset of a functionally Hausdorff space is given. Moreover, the following theorem is proved: if $X$ is a functionally Hausdorff space, then $\left|X\right|\le 2^{\chi \left(X\right)\mathit{\text{wcd}}\left(X\right)}$.

We prove that (A) if a countably compact space is the union of countably many $D$ subspaces then it is compact; (B) if a compact $T_2$ space is the union of fewer than $N\left(ℝ\right)$ = $cov\left(ℳ\right)$ left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkačenko from 1979; (A) also answers a question that was raised by Arhangel’skiǐ and improves a result of Gruenhage.