Ultrafilter invariants in topological spaces

Abstract:

Let $\mathfrak {m} \geqslant {\aleph _0}$ and $X = \prod \nolimits _{i \in I} {{X_i}}$. Then X is $[{\aleph _0}, \mathfrak {m}]$-compact if and only if $\prod \nolimits _{i \in J} {{X_i}}$ is $[{\aleph _0}, \mathfrak {m}]$-compact for all $J \subset I$ with $|J| \leqslant {2^{{2^\mathfrak {m}}}}$. Let $\mathfrak {m} \geq {\aleph _0}$, $({x_\xi }: \xi < \mathfrak {m})$ a net in X, $p \in X$, and $\mathcal {D} \in \beta (\mathfrak {m})$. Then $p = \mathcal {D} - {\lim _{\xi < \mathfrak {m}}} {x_\xi }$ if $\{ \xi < \mathfrak {m}: {x_\xi } \in U \} \in \mathcal {D}$ for every neighborhood U of p. Every topological space is characterized by its $\mathcal {D}$-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then X is a continuous perfect image of a subspace of $\beta D$ which contains D if and only if X is regular.