continuous relation

The idea of a continuous relationPlanetmathPlanetmath is neither as old nor as well-established as the idea of a continuous functionMathworldPlanetmathPlanetmath. Different authors use somewhat different definitions. The present article is based on the following definition:

Let X and Y be topological spacesMathworldPlanetmath and R a relationMathworldPlanetmath between X and Y (R is a subset of X×Y). R is said to be if

for any open subset V of Y, R-1(V) is open in X.

Here R-1(V) is the inverse image of V under R, and is defined as

R-1(V):={xXxRy for some yV}.

Equivalently, R is a continuous relation if for any open set V of Y, the set πX((X×V)R) is open in X, where πX is the projection map X×YX.

Remark. Continuous relations are generalizationPlanetmathPlanetmath of continuous functions: if a continuous relation is also a function, then it is a continuous function.

Some examples.

  • Let X be an ordered space. Then the partial orderMathworldPlanetmath is continuous iff for every open subset A of X, its lower set A is also open in X.

    In particular, in , the usual linear orderingPlanetmathPlanetmath on is continuous. To see this, let A be an open subset of . If A=, then A= as well, and so is open. Suppose now that A is non-empty and deal with the case when A is not bounded from above. If r, then there is aA such that ra, so that rA, which implies A=. Hence A is open. If A is bounded from above, then A has a supremumMathworldPlanetmathPlanetmath (since is Dedekind complete), say x. Since A is open, xA (or else x(a,b)A, implying x<x+b2(a,b), contradicting the fact that x is the least upper bound of A). So A=(-,x), 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 R={(x,y)2x2+y2=1}. This is not a continuous relation. Take A=(-2,2), which is open. But then R-1(A)=[-1,1], which is closed.

  • Now, let X be a locally connected topological space. For any x,yX, define xy iff x and y belong to the same connected componentMathworldPlanetmathPlanetmathPlanetmathPlanetmath of X. Let A be an open subset of X. Then B=-1(A) is the union of all connected components containing points of A. Since (it can be shown) each connected component is open, so is their union, and hence B is open. Thus is a continuous relation.

  • If R is symmetricPlanetmathPlanetmath, then R is continuous iff R-1 is. In particular, in a topological space X, an equivalence relationMathworldPlanetmath on X is continuous iff the projection p of X onto the quotient spaceMathworldPlanetmath X/ is an open mapping.

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 setsPlanetmathPlanetmath to be closed (making the relation upper and lower semi-continuous). Wyler suggests the following definition: If r:ef is a relation between topological spaces E and F, then r is continuous iff for each topological space A, and functions f:AE and g:AF such that f(u)𝑟g(u) for all uA, continuity of f implies continuity of g.


  • 1 T. S. Blyth, Lattices and Ordered Algebraic StructuresPlanetmathPlanetmath, Springer, New York (2005).
  • 2 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 3 Oswald Wyler, CharacterizationMathworldPlanetmath of Regularity in TopologyMathworldPlanetmath Proceedings of the American Mathematical Society, Vol. 29, No. 3. (Aug., 1971), pp. 588-590.
Title continuous relation
Canonical name ContinuousRelation
Date of creation 2013-03-22 17:05:39
Last modified on 2013-03-22 17:05:39
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 54A99