$L$-paracompactness and $L_2$-paracompactness

Abstract

A topological space $X$ is called $L$-paracompact if there exist a paracompact space $Y$ and a bijective function $f:X\longrightarrow Y$ such that the restriction $f\upharpoonright_{A}:A\longrightarrow f(A)$ is a homeomorphism for each Lindelö}f subspace $A\subseteq X$. A topological space $X$ is called $L_2$-paracompact if there exist a Hausdorff paracompact space $Y$ and a bijective function $f:X\longrightarrow Y$ such that the restriction $f\upharpoonright_{A}:A\longrightarrow f(A)$ is a homeomorphism for each Lindelöf subspace $A\subseteq X$. We investigate these two properties.