Example. On the set R of all real numbers we define a metric d by
d(x,y)=1 if x≠y, d(x,y)=0 if x=y.
Check that d is translation invariant on R but that d fails to be homogeneous w.r.t. (scalar) multiplication. The open unit ball centered at any x in (R,d) is {x} and the closed unit ball of x is R. The closure of {x} is {x}, which is different from R. The topology of (R,d) is discrete and is not homéomorphic to the vector space topology on R induced by the standard norm (or by any norm).
A metric d on a vector space X is called translation invariant if
d(x+u,y+u)=d(x,y), for all x,y,u in X.
The metric d is said to be homogeneous if
d(ax,ay)=|a| d(x,y), for all x,y in X, and for all a in the field F.
Remark. If a metric d is both translation invariant and homogeneous on X, we may define a norm by
||x||=d(x,0), for all x in X.
You might wish to verify that this really gives a norm on X, and that d(...