Fractional topological dimension

Posted in Math by Mike Stay on 2007 July 27

We say a topological space X has topological dimension m if every covering C of X has a refinement C' in which every point of X occurs in at most m+1 sets in C', and m is the smallest such.

Any point x is clearly in at least one set; it’s the other sets that x might be in that concern us. The dimension m turns up in a sum over sets in the refined covering C'. If we want to move to groupoid cardinality, then the sum should be over equivalence classes of sets in C'.

Let C'_{(x,A)} be the set \{A' \in C', A' \ne A, x \in A' \}. Given (x,A), we need some way to get a group G_{(x,A)} acting on C'_{(x,A)}. If we have such a group, then we can say the topological dimension is the maximum over (x,A) of the groupoid cardinalities of C'_{(x,A)} // G_{(x,A)}.

Any idea how to get such a group?


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