# 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.

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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.

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Recall that the Katětov extension $\kappa X$ of a Hausdorff space $X$, is the set $X$ ∪ {p:p is a free open ultrafilter on $X$} with the topology generated by {{p} ∪ U : U ∈ p ∈ $\kappa X\backslash X$} ∪ {U : U $\in \tau(X)$} ($\tau(X)$ denotes the set of open sets of $X$).

It follows from Exercise 7B in the Porter-Woods text (Extensions and Absolutes of Hausdorff Spaces) that if X is discrete, $\kappa X$ is $\beta X$ with the finer topology generated by {U : U $\in \tau(X)$} ∪ { {p} ∪ U: p ∈ cl$_{\beta X}$U, U $\in \tau(X)$}.

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