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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.

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...

In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $𝒫$, then $G$ and $bG\setminus G$ are separable and metrizable, where $𝒫$ is a class of spaces which satisfies the following conditions: (1) if $X\in 𝒫$, then every compact subset of the space $X$ is a...

For any ordinal $\lambda$ of uncountable cofinality, a $\lambda$-tree is a tree $T$ of height $\lambda$ such that $|T_{\alpha }|<\mathrm{cf}\left(\lambda \right)$ for each $\alpha <\lambda$, where $T_{\alpha }=\{x\in T:\mathrm{ht}\left(x\right)=\alpha \}$. In this note we get a Pressing Down Lemma for $\lambda$-trees and discuss some of its applications. We show that if $\eta$ is an uncountable ordinal and $T$ is a Hausdorff tree of height $\eta$ such that $|T_{\alpha }|\le \omega$ for each $\alpha <\eta$, then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C\subset T$ and for each limit ordinal $\alpha \le \eta$ with $\mathrm{cf}\left(\alpha \right)>\omega$, $\left\{\mathrm{ht}\right(c):c\in C\}\cap \alpha$ is not stationary in $\alpha$. In the last part of this note, we investigate some...

Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_{\delta }$-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\left\{x\right\}=\bigcap \left\{g\right(x,n):n\in \mathbb{N}\}$ and $g(x,n+1)\subseteq g(x,n)$ for each $n\in \mathbb{N}$. (2) If a...