# continuous relation

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

 $R^{-1}(V):=\{x\in X\mid xRy\mbox{ for some }y\in V\}.$

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 $X\times Y\to X$.

Some examples.

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 $r\colon e\to f$ is a relation between topological spaces $E$ and $F$, then $r$ is continuous iff for each topological space $A$, and functions $f\colon A\to E$ and $g\colon A\to F$ such that $f(u)\mathrel{r}g(u)$ for all $u\in A$, continuity of $f$ implies continuity of $g$.

## References

Title continuous relation ContinuousRelation 2013-03-22 17:05:39 2013-03-22 17:05:39 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 54A99