Open and Closed Relations: A Topological Approach to Evaluating Limits

In summary, the conversation discusses the concept of open and closed relations in simply ordered metric spaces. It is defined as a relation that holds for convergent sequences but not necessarily for their limits, and its closure is the strictest relation that holds for the limits. This concept is useful for evaluating limits, but the terminology varies and may be referred to as the closure of a set or a relation. It is also mentioned that in general topological spaces, working with nets is necessary for equivalence.
  • #1
alexfloo
192
0
"Open" and "closed" relations

We know that if we have convergent sequences (xn) and (yn) in simply ordered metric space, then xn[itex]\leq[/itex]yn implies that the limits x and y have x[itex]\leq[/itex]y. Also, xn<yn.

My instinct on noting this is to say that "<" is an "open relation" on that metric space, and that "[itex]\leq[/itex]" is its "closure" in that this pair of relations shares a certain property of open sets and their closures in a topological sense.

More generally, I would define an "open" relation to be a relation such that, if it holds pairwise for two convergent sequences (or equivalently if it eventually holds pairwise), it does not necessarily hold in their limits. It is "closed" if this does imply that it holds in the limit. The "closure" of a relation R is the "strictest" (in some sense I'm not certain of) relation such that if R holds pairwise on two sequences, then the closure holds for the limits.

This seems like it would be a useful concept for evaluating limits, but a Google search didn't turn up anything with my particular terminology. Is this an actual developed topic? And if so, what it the common terminology for it?
 
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  • #2


A relation R can be seen as a subset of X x X, so what you are looking for is probably the closure of R as a subset of X x X in the induced product topology from X, where X is your ordered space. It is the smallest relation (as a set) which is closed (by your definition) and contains R. Closed relations correspond to closed subsets of X x X.

I would define open relations as relations corresponding to the open subsets of X x X, so that the space of open relations form a topology on the powerset of relations.
 
  • #3


That was my intuition, but I wasn't sure about it. So it is the case that R being a closed subset of X x X is equivalent to R being a "closed relation" as above?
 
  • #4


alexfloo said:
That was my intuition, but I wasn't sure about it. So it is the case that R being a closed subset of X x X is equivalent to R being a "closed relation" as above?

On metric spaces, yes.
In general topological spaces, no. The reason is that sequences are too weak to characterize the closure in topological spaces. So you need to work with nets. If you work with nets, then they become equivalent.
 
  • #5


Thanks, I haven't really been exposed much to topologies in general, which is why I specified metric spaces in the original post. I'm not familiar with the idea of a net, but I definitely plan an looking it up. Thanks a lot, both of you!
 
  • #6


Yes, I agree with micromass, I think we require the space to be first-countable in order for this to be equivalent. A closed relation (if seen as the smallest relation that contains the limit of its sequences), is not necessarily closed for a space that is not first-countable, but the closed subsets of X x X are certainly closed relations.

The open relations if defined as the complements of the closed relations is however not necessarily a topology on the powerset of relations.

But I think that my suggestion is the "best" way to define the closure if you want the open relations to be a topology (in that it is the finest topology that can be generated in such a way).
 
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What is an open relation?

An open relation is a type of mathematical relation where there is no restriction on the elements that can be related to each other. This means that any element in the first set can be related to any element in the second set.

What is a closed relation?

A closed relation is a type of mathematical relation where there are restrictions on the elements that can be related to each other. This means that only certain elements in the first set can be related to certain elements in the second set.

What is the difference between an open and closed relation?

The main difference between an open and closed relation is the level of restriction on the elements that can be related to each other. In an open relation, there are no restrictions, while in a closed relation, there are specific restrictions on the elements that can be related.

What are some examples of open relations?

Examples of open relations include the relation "x is divisible by y," where x and y can be any integers, and the relation "x is a multiple of y," where x and y can be any real numbers.

What are some examples of closed relations?

Examples of closed relations include the relation "x is greater than y," where x and y are restricted to be positive integers, and the relation "x is a factor of y," where x and y are restricted to be prime numbers.

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