A topology that is "potentially" a metric topology, in the sense that one can define a suitable metric that induces it. The word "potentially" here means that although the metric exists, it may be unknown.
In fact, there are sufficient criteria on the topology that assure the existence of such a metric even if this is not explicitly given. An example of an existence theorem of this kind is due to Urysohn (Kelley 1955, p. 125), who proved that a regular T1-space whose topology has a countable basis is metrizable.
Conversely, a metrizable space is always 
and regular, but the condition on the basis has to be weakened
since in general, it is only true that the topology has a basis which is formed by
countably many locally finite families of open sets.
Special metrizability criteria are known for T2-spaces. A compact 
-space
is metrizable iff the set of all elements 
of 
is a zero set (Willard
1970, p. 163). The continuous image of a compact metric space in a Hausdorff
space is metrizable. (Willard 1970, p. 166). This implies in particular that
a distance can be defined on every path in a T2-space.
