Continuous Images of Cantor's Ternary Set

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 that is not the continuous image of Cantor's ternary set.