Open or closed or neither?

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In summary, the set C={(x,y)|x in Q, y in R} is neither closed nor open. It consists of an infinite union of closed sets, but this does not necessarily make it closed. The fact that every real number is the limit of a sequence of rationals shows that C is not closed. Additionally, C is not open because there are points in any ball around a point in C that are not in C. This can also be shown by considering the closure of rationals in the reals, which is all of the reals. Therefore, the set C is not closed and not open.
  • #1
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Homework Statement


Consider R^2 and the set C={(x,y)|x in Q, y in R}
Is C closed, open or neither?

The Attempt at a Solution


C consists an infinite union of closed sets (infact straight lines in the plane). Is an infinite union of closed sets not closed? If so why. C is clearly not open.
 
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  • #2
An infinite union of closed sets is not necessarily closed for the same reason an infinite intersection of open sets is not necessarily open.

If the set is closed, sequences in the set with a limit have a limit in the set. Is this true?
 
  • #3
Good point.

Any infinite continued fraction equals an irrational number

Consider an infinite sequence of finite continued fractions that converge in the limit to an irrational number. The fact that each element in the sequence is a finite fraction means it is a rational number hence on the x-axis but the limit of this sequence is an irractional number which is not in C so C is not closed.

C is not open because one can draw a ball around any point in C and not every point in the ball is in C.
 
  • #4
Yea, although it's slightly easier to just note that any real number is the limit of a sequence of rationals (say, the truncated decimal expansions). Also, you need to show that there is no ball around the point contained in C, not just that some ball isn't.
 
  • #5
So I should have said for any ball with radius r>0, there will be points in the ball not in C.
 
  • #6
If a set is closed it is its own closure. WHat is the closure (in the metric topology) of the rationals in the reals? By *definition* it is all of the reals, so the rationals are not closed. (we'll assume you can assume that there are non-rational real numbers).
 

1. What is the difference between an open, closed, and neither system?

An open system allows for the exchange of matter and energy with its surroundings, a closed system only allows for the exchange of energy, and a neither system does not allow for any exchange with its surroundings.

2. Can a system change from open to closed or vice versa?

Yes, a system can change from open to closed or vice versa depending on the conditions and interactions with its surroundings.

3. What are some examples of open, closed, and neither systems?

An example of an open system is a pot of boiling water, as it allows for heat and steam to escape. A closed system could be a sealed terrarium, as it only allows for light and heat to enter and exit. A neither system could be a perfect vacuum, as it does not allow for any exchange with its surroundings.

4. What are the advantages and disadvantages of open, closed, and neither systems?

The advantage of an open system is that it allows for the exchange of matter and energy, which can lead to growth and adaptation. The disadvantage is that it can also lead to loss of control and stability. A closed system has the advantage of maintaining control and stability, but the disadvantage of limited growth and adaptation. A neither system has the advantage of complete isolation, but the disadvantage of being unable to sustain life.

5. How do scientists determine if a system is open, closed, or neither?

Scientists determine the type of system based on its interactions with its surroundings. If there is an exchange of matter and energy, it is an open system. If there is only an exchange of energy, it is a closed system. If there is no exchange, it is a neither system.

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