In this note, we show that if for any transitive neighborhood assignment $\phi$ for $X$ there is a point-countable refinement $ℱ$ such that for any non-closed subset $A$ of $X$ there is some $V\in ℱ$ such that $|V\cap A|\ge \omega$, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wc{s}^{*}$-network then $X$ is transitively $D$, and hence if $X$ is a Hausdorff $k$-space and has a point-countable $k$-network, then $X$ is transitively $D$. We prove that if $X$ is a countably compact sequential space and has a point-countable...

A neighbourhood assignment in a space $X$ is a family $𝒪=\{O_x:x\in X\}$ of open subsets of $X$ such that $x\in O_x$ for any $x\in X$. A set $Y\subseteq X$ is a kernel of $𝒪$ if $𝒪\left(Y\right)=\bigcup \{O_x:x\in Y\}=X$. If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$-space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$-space is a $D$-space and we prove an addition...

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 $\{U_{i}i\in I\}$, 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...

A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction $r:X̃\genfrac{}{}{0pt}{}{onto}{⟶}X$ such that all fibers $r^{-1}\left(x\right)$ are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum $Y_X$ is called a continuous pseudo-fan of a compactum X if there are a point $c\in Y_X$ and a family ℱ of pseudo-arcs such that $\bigcup ℱ=Y_X$, any subcontinuum of $Y_X$ intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with...

We construct in ZFC an ultrafilter $U\in {\mathbb{N}}^{*}$ such that for every one-to-one function $f:\mathbb{N}\to \mathbb{N}$ there exists $U\in U$ with $f\left[U\right]$ in the summable ideal, i.e. the sum of reciprocals of its elements converges. This strengthens Gryzlov’s result concerning the existence of $0$-points.

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.

In this note, we introduce the concept of weakly monotonically monolithic spaces, and show that every weakly monotonically monolithic space is a $D$-space. Thus most known conclusions on $D$-spaces can be obtained by this conclusion. As a corollary, we have that if a regular space $X$ is sequential and has a point-countable $wc{s}^{*}$-network then $X$ is a $D$-space.

We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.