In this note we show the following theorem: “Let $X$ be an almost $k$-discrete space, where $k$ is a regular cardinal. Then $X$ is $k^{+}$-Baire iff it is a $k$-Baire space and every point-$k$ open cover $𝒰$ of $X$ such that $card\left(𝒰\right)\le k$ is locally-$k$ at a dense set of points.” For $k={\aleph }_0$ we obtain a well-known characterization of Baire spaces. The case $k={\aleph }_1$ is also discussed.

We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if $C_p\left(X\right)$ is Lindelöf, $Y=X\cup \left\{p\right\}$, and the point $p$ has countable character in $Y$, then $C_p\left(Y\right)$ is Lindelöf; b) if $Y$ is a cozero subspace of a Tychonoff space $X$, then $l\left(C_p{\left(Y\right)}^{\omega }\right)\le l\left(C_p{\left(X\right)}^{\omega }\right)$ and $ext\left(C_p{\left(Y\right)}^{\omega }\right)\le ext\left(C_p{\left(X\right)}^{\omega }\right)$.

We study the relation between a space $X$ satisfying certain generalized metric properties and its $n$-fold symmetric product ${ℱ}_n\left(X\right)$ satisfying the same properties. We prove that $X$ has a $\sigma$-$\left(P\right)$-property $cn$-network if and only if so does ${ℱ}_n\left(X\right)$. Moreover, if $X$ is regular then $X$ has a $\sigma$-$\left(P\right)$-property $ck$-network if and only if so does ${ℱ}_n\left(X\right)$. By these results, we obtain that $X$ is strict $\sigma$-space (strict $\aleph$-space) if and only if so is ${ℱ}_n\left(X\right)$.

The Noetherian type of topological spaces is introduced. Connections between the Noetherian type and other cardinal functions of topological spaces are obtained.

We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II...

Through the study of frame congruences, new characterizations of the paracompactness of frames are obtained.

We give sufficient and necessary conditions to be fulfilled by a filter $\Psi$ and an ideal $\Lambda$ in order that the $\Psi$-quotient space of the $\Lambda$-ideal product space preserves $T_k$-properties ($k=0,1,2,3,3\frac{1}{2}$) (“in the sense of the Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.

We show that the product of a compact, sequential $T_2$ space with an hereditarily absolutely countably compact $T_3$ space is hereditarily absolutely countably compact, and further that the product of a compact $T_2$ space of countable tightness with an hereditarily absolutely countably compact $\omega$-bounded $T_3$ space is hereditarily absolutely countably compact.