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continuous relation

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Definition
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There seem to be a several definitions of "continuous relation" flying around. Inverse images of closed sets may be required to be closed, and images of points may even be required to be compact. Is there any definition that is particularly popular? Has there been much work on this topic? To what extent do theorems about continuous functions extend to continuous relations (with various definitions)?

Well, there is huge theory about multimaps (short for multivalued maps). Multimap F:X--oY between topological spaces X,Y is nothing else then function F:X-->2^Y (some authors require some additonal properties, for example F(x) to be nonempty and compact for all x\in X, but this is not necessary). If you have relation R (subset of XxY), then you can look at it as a multimap via this one-to-one correspondence (between relations on XxY and multimaps from subspaces of X to Y):

define X_R={x\inX; \exists y such that xRy}, then define F_R:X_R--oY as follows:
F_R(x):={y\inY; xRy}.

Of course essentialy there's no difference between multimaps and relations, but sometimes it is easier to think about relation as a function.

As I said there is huge theory about multimaps. There you can talk about upper semicontinous multimaps, lower semicontinous multimaps, etc. These are good properties, but property of relation to be continous (in the sense you were refering to) is not good (in my opinion), because it is too strong.

joking

> but property of relation to be continous (in the sense you were
> refering to) is not good (in my opinion), because it is too strong.

Ups... this property is exactly lower semicontinuity for multimaps. :) I was thinking about something else.

joking

For the final definition of continuousness (applying to any morphisms of certain categories) see my article "Generalized Continuousness" at
http://www.mathematics21.org/binaries/continuousness.pdf

Actually, to completely understand that article you need reading some other prerequisites from Algebraic General Topology
http://www.mathematics21.org/algebraic-general-topology.html

For multivalued functions (well, multivalued morphisms in general) there are three distinct definitions of continuousness.

P.S. Nominate me for Abel Prize - http://www.mathematics21.org/abel-prize.html
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts

It looks like I introduced an error in the LaTeX, but I can't find it!

And removed the error. Unfortunately along with the bibliography entry I was trying to add. Let's see if I can make this happen.

All better. Now all that's left is adding proper links and a proper footnote or whatever to reference my source.

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