A topological space $X$ is said to be a Menger space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and $\cup_{n\in\mathbb{N}}\mathcal{V}_n$ is an open cover of $X$. H...

A topological space $X$ is said to be a Menger space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and $\cup_{n\in\mathbb{N}}\mathcal{V}_n$ is an open cover of $X$. H...

A topological space $X$ is said to be a Hurewicz space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and each $x\in X$ belongs to $\cup\mathcal{V}_n$ for all but fini...

We say that topological space $X$ is Rothberger $S_1(\mathcal O,\mathcal O)$, if: For any sequence of open covers of $X$, $\{ \mathcal U_n | n \in \omega \}$, one can always find a sequence $\{ U_n | n \in \omega \}$, such that for each $n \in \omega$ $\{ U_n \in \mathcal U_n \}$ and $...

I have encountered the following property in this article: We say that a space $X$ parR (partition-Rothberger), if, for every sequence $(\mathcal P_n : n \in \omega)$, of partition of $X$ into clopen sets, one can pick $V_n \in \mathcal P_n$, so that $\{ V_n : n \in \omega \}$ covers $X$. a...

Definition: An $\omega$-cover of a topological space $X$, is an open cover $\mathcal U$, such that, for any finite set $C \subset X$, there exists an open set $U \in \mathcal U$, such that, $C \subset U$. Let $X=[0,\omega_1]$, where $\omega_1$ is the first uncountable ordinal. Let $\langle \mat...