Need hands to the Rational Set

In summary, Ka Yan is asking if a non-empty proper subset of the rational numbers can be both open and closed. The answer is yes, as every set of rational numbers with the usual topology is both open and closed. However, no interval of rational numbers is connected. Ka Yan apologizes for not expressing the question clearly and mentions that they are a beginner in math and English as a foreign language. They also mention that they are learning basic topology in general analysis.
  • #1
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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:smile:
 
Last edited:
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  • #2
What are your thoughts?
 
  • #3
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?
 
  • #4
what does it mean to be closed in a space that is not metric?
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
halls you didn't answer my question
 
  • #9
"Open" and "closed' are adjectives that can be applied to subsets of any topological space. (Metric spaces are a subclass of topological spaces.)
 
  • #10
ice109 said:
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.
 

1. What is the "Need hands to the Rational Set" theory?

The "Need hands to the Rational Set" theory is a concept in rationality that suggests that in order for something to be considered rational, it must be able to be carried out through physical actions or manipulations. In other words, rationality requires the use of hands as a means of problem-solving or decision-making.

2. Why is the use of hands important in rationality?

The use of hands plays a crucial role in rationality because it allows for the execution of logical and analytical thinking. Hands are essential tools for manipulating objects, carrying out experiments, and communicating ideas through writing or gestures. Without hands, our ability to think and reason would be severely limited.

3. Can rationality be achieved without the use of hands?

While it is possible for individuals to think and make decisions without the use of hands, the "Need hands to the Rational Set" theory suggests that true rationality is not achievable without the ability to physically carry out actions. This is because hands allow us to test hypotheses, gather data, and make practical applications of our thoughts and ideas.

4. How does the "Need hands to the Rational Set" theory relate to scientific research?

In scientific research, the use of hands is crucial for conducting experiments, collecting data, and analyzing results. Without hands, scientists would not be able to manipulate equipment or perform complex procedures, making it difficult to advance our understanding of the world through empirical evidence and rational analysis.

5. Are there any criticisms of the "Need hands to the Rational Set" theory?

Some critics argue that the theory is too limiting and excludes individuals with physical disabilities or impairments that prevent them from using their hands. Others argue that rationality is a cognitive process that can be achieved without the use of hands. However, proponents of the theory suggest that while rationality may not be impossible without hands, it is still an essential factor in fully realizing and expressing our rational capabilities.

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