# What is Katetov extension of the natural numbers?

I read a paper and met the concept Katetov extension. What is Katetov extension of the natural numbers? Reference on it are also welcome.

-
What paper?${}$ –  Chris Eagle Dec 23 '12 at 13:28

In

M. Katětov, Über H-abgeschlossene und bikompakte Räume, Cas. Mat. Fys., 69:39-49, 1940,

Katětov proved that any Hausdorff space $X$ can be densely embedded in an $H$-closed space; he did it by constructing a specific $H$-closed extension, $\kappa X$, of $X$. (A space is $H$-closed if it is closed in every Hausdorff space in which it is embedded.)

Definition. If $\langle X,\tau\rangle$ is a Hausdorff space, $\kappa X$, the Katětov extension of $X$, is the set $$X\cup\{p\subseteq\tau:p\text{ is a free open ultrafilter on }X\}$$ with the topology generated by $$\big\{\{p\}\cup U:U\in p\in\kappa X\setminus X\big\}\;.$$

The first paper here, by Mukherjee, Sengupta, and Ghosh, treats some cardinal functions on $\kappa D$ for discrete spaces $D$.

Added: That link goes to a front page from which it’s not obvious how to reach the paper. Here is a direct link to the PDF. Katětov’s original paper is freely available here.

-