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.
Read more about this topic: Continuous Function
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