# Supercompact minus compact is super

## TL;DR

This paper introduces the concept of super spaces characterized by binary closed $k$-networks, expanding the class of spaces known to be supercompact and showing their closure properties under various topological operations.

## Contribution

It defines super spaces via binary closed $k$-networks and proves they include all GO-spaces, supercompact, metrizable, and collectionwise normal $eth$-spaces, with closure under Tychonoff products.

## Key findings

- Super spaces include all GO-spaces and supercompact spaces.
- The class of super spaces is closed under Tychonoff products.
- Metrizable spaces are super, implying all compact metrizable spaces are supercompact.

## Abstract

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed $k$-network. This characterization suggests to call a topological space $super$ if it has a binary closed $k$-network $\mathcal N$. The binarity of $\mathcal N$ means that every linked subfamily of $\mathcal N$ is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal $\aleph$-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szyma\'nski (1975) and then reproved by Mills (1979), van Douwen (1981), and D{\c e}bski (1984).