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 $\mathcal {P}$, then $G$ and $bG\setminus G$ are separable and metrizable, where $\mathcal {P}$ is a class of spaces which satisfies the following conditions: (1) if $X\in \mathcal {P}$, then every compact subset of the space $X$ is a $G_\delta $-set of $X$; (2) if $X\in \mathcal {P}$ and $X$ is not locally compact, then $X$ is not locally countably compact; (3) if $X\in \mathcal {P}$ and $X$ is a Lindelöf $p$-space, then $X$ is metrizable. Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group $G$ has a compactification $bG$ such that compact subsets of $bG\setminus G$ are $G_{\delta }$-sets in a uniform way (i.e., $bG\setminus G$ is CSS), then $G$ and $bG\setminus G$ are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ has a point-countable weak base and has a dense subset $D$ such that every point of the set $D$ has countable pseudo-character in the remainder $bG\setminus G$ (or the subspace $D$ has countable $\pi $-character), then $G$ and $bG\setminus G$ are both separable and metrizable.
In this note, we show that if for any transitive neighborhood assignment $\phi $ for $X$ there is a point-countable refinement ${\mathcal F}$ such that for any non-closed subset $A$ of $X$ there is some $V\in {\mathcal F}$ such that $|V\cap A|\geq \omega $, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wcs^*$-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 $wcs^*$-network, then $X$ is compact. We point out that every discretely Lindelöf space is transitively $D$. Let $(X, \tau )$ be a space and let $(X, {\mathcal T})$ be a butterfly space over $(X, \tau )$. If $(X, \tau )$ is Fréchet and has a point-countable $wcs^*$-network (or is a hereditarily meta-Lindelöf space), then $(X, {\mathcal T})$ is a transitively $D$-space.
For any ordinal $\lambda $ of uncountable cofinality, a $\lambda $-tree is a tree $T$ of height $\lambda $ such that $|T_{\alpha }|<{\rm cf}(\lambda )$ for each $\alpha <\lambda $, where $T_{\alpha }=\{x\in T\colon {\rm ht}(x)=\alpha \}$. 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 }|\leq \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 \leq \eta $ with ${\rm cf}(\alpha )>\omega $, $\{{\rm ht}(c)\colon c\in C\} \cap \alpha $ is not stationary in $\alpha $. In the last part of this note, we investigate some properties of $\kappa $-trees, $\kappa $-Suslin trees and almost $\kappa $-Suslin trees, where $\kappa $ is an uncountable regular cardinal.