On $\kappa$-bounded and $M$-compact reflections of topological spaces

Abstract

For a topological space $X$ its reflection in a class $\mathsf T$ of topological spaces is a pair $(\mathsf T X,i_X)$ consisting of a space $\mathsf T X\in\mathsf T$ and continuous map $i_X:X\to \mathsf T X$ such that for any continuous map $f:X\to Y$ to a space $Y\in\mathsf T$ there exists a unique continuous map $\bar f:\mathsf T X\to Y$ such that $f=\bar f\circ i_X$. In this paper for an infinite cardinal $\kappa$ and a nonempty set $M$ of ultrafilters on $\kappa$, we study the reflections of topological spaces in the classes $\mathsf H_\kappa$ of $\kappa$-bounded Hausdorff spaces and $\mathsf H_M$ of $M$-compact Hausdorff spaces (a topological space $X$ is $\kappa$-bounded if the closures of subsets of cardinality $\le\kappa$ in $X$ are compact; $X$ is $M$-compact if any function $x:\kappa\to X$ has a $p$-limit in $M$ for every ultrafilter $p\in M$).