Metrizable spaces A topological space X, is metrizable if it is homeomorphic to a metric space.
I want to know, does this mean that all of topological properties of a metric space, inherit to that topological space? Also I'm asking, does all the importance of metrizability, is the inheritance of the topological property, or it has to be beyond that?
 A: A metric space $(X,d)$ has a standard topology $\mathcal{T}_d$ associated with it, generated by the open balls in the $d$-metric. This topology always has special properties, that general topologies do not always have:


*

*$(X,\mathcal{T}_d)$ is always first countable.

*$(X,\mathcal{T}_d)$ is always perfectly normal (including $T_1$).

*$(X,\mathcal{T}_d)$ is compact iff it is countably compact iff it is pseudocompact.

*$(X,\mathcal{T}_d)$ is second countable iff it is separable iff it is ccc.

*$(X,\mathcal{T}_d)$ is paracompact and monotonically normal.


etc etc. So if we know that a space $(X,\mathcal{T})$ is metrizable, i.e. we know there exist some metric $d$ on $X$ such that $\mathcal{T}_d = \mathcal{T}$, or equivalently that $X$ is homeomorphic to a metrizable space (we then transport the metric on the homeomorphic space to $X$ using the bijection etc.), then we know that $(X, \mathcal{T})$ also satisfies the above list of properties. The $d$ is by no means unique, but the fact that it exists, means that the space satisfies all of the above properties (which are topological, and so are preserved by homeomorphisms).
There has been a lot of research on finding criteria on the topology on $X$ to know that it is metrizable, and many so-called metrication theorems are known now. It's not a very active area of research any more. But the theorems are widely applicable, and it's very useful to know a space is metrizable because of the extra structure this gives us.  
