Continuous Function - Related Notions

Related Notions

Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in order theory, an order-preserving function f: X → Y between two complete lattices X and Y (particular types of partially ordered sets) is continuous if for each subset A of X, we have sup(f(A)) = f(sup(A)). Here sup is the supremum with respect to the orderings in X and Y, respectively. Applying this to the complete lattice consisting of the open subsets of a topological space, this gives back the notion of continuity for maps between topological spaces.

In category theory, a functor

between two categories is called continuous, if it commutes with small limits. That is to say,

for any small (i.e., indexed by a set I, as opposed to a class) diagram of objects in .

A continuity space is a generalization of metric spaces and posets, which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.