Continuous images of Cantor's ternary set

TL;DR

This paper explores the limitations of Cantor's ternary set as a universal continuous image by constructing a specific compact, countably infinite, non-Hausdorff space that cannot be obtained as its continuous image.

Contribution

It provides a counterexample of a compact, countably infinite, non-Hausdorff space that is not a continuous image of Cantor's ternary set, extending the understanding of the theorem's scope.

Findings

Counterexample of a non-Hausdorff space not image of C

Shows limitations of the Hausdorff condition in the theorem

Highlights differences between Hausdorff and non-Hausdorff spaces

Abstract

The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set $C$. It is well known that there are compact Hausdorff spaces of cardinality equal to that of $C$ that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of $C$. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.