Fractional topological dimension
We say a topological space has topological dimension if every covering of has a refinement in which every point of occurs in at most sets in , and is the smallest such.
Any point is clearly in at least one set; it’s the other sets that might be in that concern us. The dimension turns up in a sum over sets in the refined covering . If we want to move to groupoid cardinality, then the sum should be over equivalence classes of sets in .
Let be the set . Given , we need some way to get a group acting on . If we have such a group, then we can say the topological dimension is the maximum over of the groupoid cardinalities of
Any idea how to get such a group?