what is the relation between the Physics Density and the Topological Density

Question

A relation between both is that both deal with how accumulate is something (the notion of density per se), however, I was wondering if there is any "math" relation between both?

Thanks

Answer 1

In a topological space $X$, a set $A\subset X$ is dense if $\forall x\in X$ any neighborhood of $x$ contains at least one point of $A$.

In the physical sense: given a specified volume and a mass: $$ \text{density}=\frac{\text{mass}}{\text{volume}}.$$

We do not have a notion of volume/distance in a general topological space, nor do we have a notion of mass. So, there is no way to evaluate the physical definition of density in a topological space. However, you can attempt to relate the two by considering that the topological notion of density attempts to create an idea of $A$ being packed in tightly into $X$.

If you imagine that the set $X$ has a certain "volume", and the set $A$ has a certain "mass", then $A$ is dense in $X$ if there's a large mass of $A$ in $X$'s volume.

In conclusion, the two are not really related mathematically- but the topological notion attempts to provoke an idea of physical density.