continuous relation
The idea of a continuous relation^{} is neither as old nor as wellestablished as the idea of a continuous function^{}. Different authors use somewhat different definitions. The present article is based on the following definition:
Let $X$ and $Y$ be topological spaces^{} and $R$ a relation^{} between $X$ and $Y$ ($R$ is a subset of $X\times Y$). $R$ 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
$${R}^{1}(V):=\{x\in X\mid xRy\text{for some}y\in V\}.$$ 
Equivalently, $R$ is a continuous relation if for any open set $V$ of $Y$, the set ${\pi}_{X}((X\times V)\cap R)$ is open in $X$, where ${\pi}_{X}$ is the projection map $X\times Y\to X$.
Remark. Continuous relations are generalization^{} 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 order^{} $\le $ is continuous iff for every open subset $A$ of $X$, its lower set $\downarrow A$ is also open in $X$.
In particular, in $\mathbb{R}$, the usual linear ordering^{} $\le $ on $\mathbb{R}$ is continuous. To see this, let $A$ be an open subset of $\mathbb{R}$. If $A=\mathrm{\varnothing}$, then $\downarrow A=\mathrm{\varnothing}$ as well, and so is open. Suppose now that $A$ is nonempty and deal with the case when $A$ is not bounded from above. If $r\in \mathbb{R}$, then there is $a\in A$ such that $r\le a$, so that $r\in \downarrow A$, which implies $\downarrow A=\mathbb{R}$. Hence $\downarrow A$ is open. If $A$ is bounded from above, then $A$ has a supremum^{} (since $\mathbb{R}$ is Dedekind complete), say $x$. Since $A$ is open, $x\notin A$ (or else $x\in (a,b)\subseteq A$, implying $$, contradicting the fact that $x$ is the least upper bound of $A$). So $\downarrow A=(\mathrm{\infty},x)$, which is open also. Therefore, $\le $ is a continuous relation on $\mathbb{R}$.

•
Again, we look at the space $\mathbb{R}$ with its usual interval topology. The relation this time is $R=\{(x,y)\in {\mathbb{R}}^{2}\mid {x}^{2}+{y}^{2}=1\}$. This is not a continuous relation. Take $A=(2,2)$, which is open. But then ${R}^{1}(A)=[1,1]$, which is closed.

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Now, let $X$ be a locally connected topological space. For any $x,y\in X$, define $x\sim y$ iff $x$ and $y$ belong to the same connected component^{} of $X$. Let $A$ be an open subset of $X$. Then $B={\sim}^{1}(A)$ 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 $\sim $ is a continuous relation.

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If $R$ is symmetric^{}, then $R$ is continuous iff ${R}^{1}$ is. In particular, in a topological space $X$, an equivalence relation^{} $\sim $ on $X$ is continuous iff the projection $p$ of $X$ onto the quotient space^{} $X/\sim $ 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 sets^{} to be closed (making the relation upper and lower semicontinuous). Wyler suggests the following definition: If $r:e\to f$ is a relation between topological spaces $E$ and $F$, then $r$ is continuous iff for each topological space $A$, and functions $f:A\to E$ and $g:A\to F$ such that $f(u)\mathit{r}g(u)$ for all $u\in A$, continuity of $f$ implies continuity of $g$.
References
 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=00029939%28197108%2929%3A3%3C588%3AACORIT%3E2.0.CO%3B23A Characterization^{} of Regularity in Topology^{} Proceedings of the American Mathematical Society, Vol. 29, No. 3. (Aug., 1971), pp. 588590.
Title  continuous relation 

Canonical name  ContinuousRelation 
Date of creation  20130322 17:05:39 
Last modified on  20130322 17:05:39 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  12 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54A99 