Metric on the Set of Binary rectangular matrices

Question

Consider a set of all possible Binary rectangular matrices.

How many non-equivalent metrics can be defined? How to define non equivalent metrics on this set precisely?

Answer 1

If the space consists of binary $m\times n$ matrices for fixed integers $m,n \gt 0$, then the number of such matrices is finite. Any finite metric space has a discrete topology, so there is only one metric topology up to topological equivalence.

Indeed strong equivalence of metrics must also hold in this finite setting. Let $d_1(x,y)$ and $d_2(x,y)$ be two metrics on a finite set $S$. Let $e_1,e_2$ be the minimum distances $d_1(x,y)$, resp. $d_2(x,y)$, over all distinct $x,y \in S$. Similarly let $f_1,f_2$ be the maximum distances $d_1(x,y)$, resp. $d_2(x,y)$, over all distinct $x,y \in S$. Then for any distinct $x,y \in S$:

$$ d_1(x,y) \lt (f_1 + 1) = e_2^{-1} (f_1 + 1) e_2 \le e_2^{-1} (f_1 + 1) d_2(x,y) $$

and by a similar estimate, $d_2(x,y) \lt e_1^{-1} (f_2 + 1) d_1(x,y)$. Thus the two metrics are strongly equivalent.