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 $p\in M$).

17 pages