Certain observations on local properties of topological spaces

Let $\mathcal{P}$ be any topological property of a space $X$. We say that $X$ is $\mathcal{P}$ at $x\in X$ if there exist an open set $U$ and a subspace $Y$ of $X$ satisfying $\mathcal{P}$ such that $x\in U\subseteq Y$. We also say that $X$ is locally $\mathcal{P}$ if $X$ is $\mathcal{P}$ at every point of $X$. We study this local property and obtain the following results under certain topological assumptions on $\mathcal{P}$. (1) Every locally $\mathcal{P}$ Hausdorff $P$-space can be densely embedded in a $\mathcal{P}$ Hausdorff $P$-space. (2) If a Hausdorff $P$-space $X$ is $\mathcal{P}$ at $x\in X$, then $\chi(x,X)\leq\psi(x,X)^\omega$. (3) For a locally $\mathcal{P}$ Hausdorff $P$-space $X$, $w(X)\leq nw(X)^\omega\leq |X|^\omega$. Besides, few separation like properties are obtained and preservation under certain topological operations are also investigated. Finally we present certain observations on remainders of locally $\mathcal{P}$ spaces.