While glancing over measure theory books I noticed a discrepancy in the definition of a Polish space: given a topological space $(X,\mathcal T)$, some authors use
Definition A: $X$ is a Polish space when $X$ is separable, metrizable by a distance $d$, and $(X,d)$ is complete.
Others use
Definition B: $X$ is a Polish space when $X$ is homeomorphic to a complete separable metric space.
A clearly implies B (identity is a homeomorphism).
If $(Y,d)$ is a complete separable metric space and $f:X\to Y$ is a homeomorphism, then $D:(x_1,x_2)\mapsto d(f(x_1),f(x_2))$ is a distance on $X$, and $(X,D)$ is isometric to $(Y,d)$ hence complete, and the topology on $X$ is metrized by $D$.
Why is Definition B preferred in some references ?
Best Answer
Beeing metrizable is equivalent to beeing homeomorphic to a metric space, it is even its definition sometimes. I guess that as often, it is more handy not to directly impose properties on an object X but rather to impose it on an isomorph object Y (and sometimes, Y can just be X). In that case, it seems more natural to say that a space X is homeomorphic to a well-known metric space than to construct an ad hoc distance on it as you ended up doing to prove the equivalence.