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...

We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff...

On the set $ℝ$ of real numbers we consider a poset ${𝒫}_{\tau }\left(ℝ\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq ℝ$, such that $A_1\supseteq A_2$ iff $\tau \left(A_1\right)\subseteq \tau \left(A_2\right)$. The poset has the minimal element $\tau \left(ℝ\right)$, the Euclidean topology, and the maximal element $\tau \left(\varnothing \right)$, the Sorgenfrey topology. We are interested when two topologies ${\tau }_1$ and ${\tau }_2$ (especially, for ${\tau }_2=\tau \left(\varnothing \right)$) from the poset define homeomorphic spaces $(ℝ,{\tau }_1)$ and $(ℝ,{\tau }_2)$. In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ,\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(ℝ,\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common...

Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\mathbb{N}$ is a Lindelöf space, (2) $\mathbb{Q}$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:ℝ\to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous...