Need hands to the Rational Set

[Ka Yan]

I'm a new comer, even in Math. I need hands for this (simple, may be for most people) question:

Can a subset of Rational Number Set be open (and closed)? If does, how can it be? If not,why?

thks!

Ka Yan

 

[morphism]

What are your thoughts?

 

[HallsofIvy]

What topology are you using on the rational numbers? Also, I think you mean, not "can a subset be open (and closed)" which would imply that you are really asking two questions, but "can a subset be both open and closed". That has, by the way, a trivial answer: in any topological space, the empty set and the entire set are both open and closed. What you are really asking is "can a non-empty proper subset of the rational numbers be both open and closed".

If you were to ask the same question about the real numbers, what answer would you give? What consequences would a "yes" answer have?

 

[ice109]

what does it mean to be closed in a space that is not metric?

 

[Ka Yan]

I'm sorry for not have discribed the question more clearly.

What I really wanted to ask is that, can a non-empty proper subset of the rational numbers be either open or closed ? If that subset is possiblly be either open or closed, then how can it make sense (i.e.,if possible, please discrib it in a more explicit way)? If a non-empty subset of rationals cannot be either open or closed, please explain the reason.

I have to appologize that my expression does not well enough since I'm not a EFL user.

Again thank you.

 

[HallsofIvy]

Since you did not answer my question before, I am going to assume the "usual topology" on the rational numbers- the metric topology defined by d(a, b)= |a- b|. And you are asking if it is possible for sets to be open or closed? Every topological space has open and closed sets! In fact, it is not terribly difficult to prove that every set of rational numbers, with the usual topology, is both open and closed. That was what I was referring to above. No interval of rational numbers is connected.

 

[Ka Yan]

I got you. Thank you professor!

In fact, what I'm learning now,is just basic topology within general analysis, and that's why I didnot answer your early-mentioned question. And not until now do I understand what you meant by asking that.

 

[ice109]

halls you didn't answer my question

 

[morphism]

"Open" and "closed' are adjectives that can be applied to subsets of any topological space. (Metric spaces are a subclass of topological spaces.)

 

[CRGreathouse]

what does it mean to be closed in a space that is not metric?

A topological space is defined as a set and a collection of subsets which are open. (Certain conditions apply to the collection.) So closure in a topology means that the complement of the set in question is in the collection of open sets.