Nice set theoretic formalization of the concept of convergence to points.
Metric spaces are topological spaces. But to put a metric on a blank topological space, you first have to make it a uniform space and then be lucky that it has a countable basis. Lots of important topological spaces which cannot be endowed with a metric can be found amoung the locally convex vector spaces, e.g. the space of generalized functions from the theory of partial differential equations.

A topological space is a set of points X, and a set O of subsets of X. Elements of O are called open sets. O must satisfy that finite intersections and any unions of open sets are also open sets; the empty set and the entire space, X, must also be open sets. Complements of open sets are called closed sets.

Any metric space defines a topological space by taking the topology of open spheres: for every point x and every positive r, place {y | d(x,y)<r} in O; then add all other necessary sets (all unions of finite intersections of spheres). Many topological spaces are not metrisable (no metric on their set of points yields their topology), but many concepts of metric spaces which rely only on their continuity transfer easily to a topological wording. Replacing the ubiquitous references to "open sphere"s with uses of generalised open sets allows topology to discuss continuous functions even where no metric exists. Topology really only distinguishes topological spaces up to homeomorphism.

Someone should really node some examples of metric spaces.

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