Point-set topology is the study of the general abstract nature of continuity on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness.

The low-level language of topology, which is not really considered a separate "branch" of topology. Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness. The intermediate value theorem (which states that if a path in the real line connects two numbers, then it passes over every point between the two) is a basic topological result. Others are that Euclidean n-space is homeomorphic to Euclidean m-space iff m = n, and that real valued functions achieve maxima and minima on compact sets.

algebraic topology | differential topology | low-dimensional topology | topology

college level