What is the difference between topological and metric spaces? What is the difference between a topological and a metric space?

A metric space gives rise to  a topological space on the same set (generated by the open balls in the metric). Different metrics can give the same topology. A topology that arises in this way is a metrizable topology. Using the topology we can define notions that are purely topological, like convergence, compactness, continuity, connectedness, dimension etc. Using the metric we can talk about other things that are more specific to metric spaces, like uniform continuity, uniform convergence and stuff like Hausdorff dimension, completeness etc, and other notions that do depend on the metric we choose. Different metrics that yield the same topology on a set can induce different notions of Cauchy sequences, e.g., so that the space is complete in one metric, but not in the other. In analysis e.g. one often is interested in both of these types of notions, while in topology only the purely topological notions are studied. In topology we can in fact characterize those topologies that are induced from metrics. Such topologies are quite special in the realm of all topological spaces. So in short: all metric spaces are also topological spaces, but certainly not vice versa.
