Incomparable Metrics on the Cantor Space

Publication Date


Type of Culminating Activity


Degree Title

Master of Science in Mathematics



Major Advisor

Stefan Geschke


A Cantor Space is any topological space that is homeomorphic to the Cantor Set. Cantor Spaces are precisely the totally disconnected compact metric spaces that have no isolated points. Hence, Cantor Spaces are natural candidates for recent studies including pair-colorings [3] and homeomorphic product measures [2]. Our investigation aims to produce uncountably many substantially different ways to measure distance between elements of the Cantor Set (i.e. uncountably many metrics). When we say "substantially different" , we mean that these metrics cannot be compared by scaling one above another (see definitions on following pages). We also restrict our attention to metrics that induce the usual topology on the Cantor Set, which is obtained from intersecting open sets of the real line with the Cantor Set.

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