DefinitionContinuity and compactness are interesting properties of topological spaces that have to do with how functions can be defined within the spaces and with the behaviour of sets in these spaces. Continuity preserves continuity and points closed to each other under a map, but compactness implies a set is closed and bounded loving finite open covers. These ideas are foundational of topology, analysis, and geometry to the approach continuity, and convergence of sequences and functions. I.e., the function f(x)=x1 is continuous on the interval (0,∞), but not so on a compact domain [0,∞) even though it is bound.
