# reperiendi

## 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?