The idea of a continuous relation is neither as old nor as well-established as the idea of a continuous function. Different authors use somewhat different definitions. The present article is based on the following definition:
for any open subset of , is open in .
Here is the inverse image of under , and is defined as
Equivalently, is a continuous relation if for any open set of , the set is open in , where is the projection map .
In particular, in , the usual linear ordering on is continuous. To see this, let be an open subset of . If , then as well, and so is open. Suppose now that is non-empty and deal with the case when is not bounded from above. If , then there is such that , so that , which implies . Hence is open. If is bounded from above, then has a supremum (since is Dedekind complete), say . Since is open, (or else , implying , contradicting the fact that is the least upper bound of ). So , which is open also. Therefore, is a continuous relation on .
Again, we look at the space with its usual interval topology. The relation this time is . This is not a continuous relation. Take , which is open. But then , which is closed.
Now, let be a locally connected topological space. For any , define iff and belong to the same connected component of . Let be an open subset of . Then is the union of all connected components containing points of . Since (it can be shown) each connected component is open, so is their union, and hence is open. Thus is a continuous relation.
Remark. Alternative definitions: One apparently common definition (as described by Wyler) is to require inverse images of open sets to be open and inverse images of closed sets to be closed (making the relation upper and lower semi-continuous). Wyler suggests the following definition: If is a relation between topological spaces and , then is continuous iff for each topological space , and functions and such that for all , continuity of implies continuity of .
- 1 T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
- 2 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 3 Oswald Wyler, http://links.jstor.org/sici?sici=0002-9939%28197108%2929%3A3%3C588%3AACORIT%3E2.0.CO%3B2-3A Characterization of Regularity in Topology Proceedings of the American Mathematical Society, Vol. 29, No. 3. (Aug., 1971), pp. 588-590.