A note on base-paracompact and monotone base-covering properties

Abstract

In the first part of this note, we show that if X is  a paracompact Hausdorff space and there is a locally compact closed subspace Y of  X such that for every x∈X\ Y there exists an open neighborhood O_x of x in X such that the closure of O_x is base-paracompact, then the space X is base-paracompact. In the second part of this note, we introduce notions of monotonically base-paracompact (monotonically base-metacompact, monotonically base-Lindel\"{o}f) spaces and discuss some of their properties.