Meaning of topological distance between 2 pixels
Nonlinear Signal and Image Processing: Theory, Methods, and Applications defines topological distance as follows. First, you have to define when two pixels are neighbors. The book offers two possibilities:


The neighbors of a pixel are the 4 "cardinal" pixels around it.
The neighbors of a pixel are the 8 pixels surrounding it.


This defines a graph (two pixels are connected by an edge if they are neighbors). The topological distance between two pixels $p,q$ is then the graph distance between $p$ and $q$.

The first choice above corresponds to $L^1$ ("Manhattan") distance, and the second choice to $L^\infty$ distance. This contrasts with the usual $L^2$ (Euclidean) distance.