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...