### Semi-radiality in products.

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We provide a further estimate on the cardinality of a power homogeneous space. In particular we show the consistency of the formula $\left|X\right|\le {2}^{\pi \chi \left(X\right)}$ for any regular power homogeneous ccc space.

It is proved that the product of two pseudo radial compact spaces is pseudo radial provided that one of them is monolithic.

A sufficient condition that the product of two compact spaces has the property of weak approximation by points (briefly WAP) is given. It follows that the product of the unit interval with a compact WAP space is also a WAP space.

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

We prove a dichotomy theorem for remainders in compactifications of homogeneous spaces: given a homogeneous space $X$, every remainder of $X$ is either realcompact and meager or Baire. In addition we show that two other recent dichotomy theorems for remainders of topological groups due to Arhangel’skii cannot be extended to homogeneous spaces.

A topological space is KC when every compact set is closed and SC when every convergent sequence together with its limit is closed. We present a complete description of KC-closed, SC-closed and SC minimal spaces. We also discuss the behaviour of the finite derived set property in these classes.

In questa Nota, dati uno spazio metrico perfetto $X$ ed un suo sottoinsieme $K$ chiuso e raro, si dimostra l'esistenza di una funzione continua $f:X\to [0,1]$ tale che $int({f}^{-1}(t))=\mathrm{\varnothing}$ per ogni $t\in [0,1]$, $f(x)=0$ per ogni $x\in K$ e $f(y)=1$ per qualche $y\in X\setminus K$. In particolare, ciò permette di dare risposta simultaneamente a due questioni poste in [2]. Si mettono in evidenza, poi, ulteriori conseguenze di tale risultato.

An estimate for the Novak number of a hyperspace with the Vietoris topology is given. As a consequence it is shown that this cardinal function can decrease passing from a space to its hyperspace.

A sufficient condition for the pseudo radiality of the product of two compact Hausdorff spaces is given.

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between H-closed and minimal Hausdorff spaces. An H-closed space $X$ is produced with the properties that $\left|X\right|>{2}^{{2}^{\psi \left(X\right)}}$ and $\overline{\psi}\left(X\right)>{2}^{\psi \left(X\right)}$.

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.

Countable tightness is compared to the stronger notion of countable fan-tightness. In particular, we prove that countable tightness is equivalent to countable fan-tightness in countably compact regular spaces, and that countable fan-tightness is preserved by pseudo-open compact mappings. We also discuss the behaviour of countable tightness and of countable fan-tightness under the product operation.

Several remarks on the properties of approximation by points (AP) and weak approximation by points (WAP) are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: (a) There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. (b) ${C}_{p}$ over $\sigma $-compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while...

A DC-space (or space of dense constancies) is a Tychonoff space $X$ such that for each $f\in C\left(X\right)$ there is a family of open sets $\{{U}_{i}\phantom{\rule{0.222222em}{0ex}}i\in I\}$, the union of which is dense in $X$, such that $f$, restricted to each ${U}_{i}$, is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean $f$-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions...

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