Continuous images of Cantor's ternary set
Fabian Dreher, Tony Samuel

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 . It is well known that there are compact Hausdorff spaces of cardinality equal to that of 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 . Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.
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