On a certain local version of the Rothberger property

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\begin{abstract} We study a local variant of the Rothberger property, namely the locally Rothberger property. A space $X$ is said to have the Rothberger property at $x\in X$ if there is an open set $U$ and a Rothberger subspace $Y$ of $X$ such that $x\in U\subseteq Y$. A space $X$ has the locally Rothberger property (or, $X$ is locally Rothberger) if $X$ is Rothberger at every point of $X$. The following results are obtained. \begin{enumerate} \item $\non(\text{locally Rothberger})=\cov(\mathcal{M})$. \item Every locally Rothberger Hausdorff $P$-space can be densely embedded in a Rothberger Hausdorff $P$-space. \item For a locally Rothberger Hausdorff $P$-space $X$, $w(X)\leq nw(X)^\omega$. \item If a Hausdorff $P$-space $X$ is Rothberger at $x\in X$, then the character $\chi(x,X)$ of $x$ is the smallest cardinal number of the form $|\mathcal{B}|^\omega$, where $\mathcal{B}$ is a family of open subsets of $X$ such that $\cap\mathcal{B}=\{x\}$. \end{enumerate} Besides, few separation type properties are obtained and preservation under ceratin topological operations are also investigated. Finally we present certain observations on remainders of the local variant of the Rothberger property. \end{abstract}