Which sets are open, closed, and compact?

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In summary: The integers are a discrete subset of R. The integers are closed in R, but not open and not compact. The set R-Z is open in R, but not closed, and not compact. The set R-Z has no accumulation points in R, because the set is already the whole space R, so there is no point in R that is not in R-Z that is a limit point of R-Z in R. It should be stated, as I said before, that the set of integers Z is closed in R, because the complement is a union of open balls with integer centers. The set Z is not compact, because the set has an open cover with no finite subcover. The set R-Z is not closed in R, but
  • #1
llursweetiell
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Homework Statement


Can someone check this for me?
Problem: determine which, if any of the sets if open? closed? compact?
R=reals; Q=rationals and Z=integers.
A= [0,1) U (1,2) is NEITHER
B=Z is CLOSED
C=(.5,1) U (.25,.5) U (.125, 25) U... is OPEN
D={r*sqrt(2) such that r is an element of Q} is NEITHER
E=R-Z is OPEN


Homework Equations


How do you find the set of accumuluation points for each of the above sets?

The Attempt at a Solution


my reasoning:
A) the union is [0,1) -1. And [0,1) is neither open nor closed and 1 is bounded. so A is neither.
B) closed by definition?
C) union of open intervals is open
D) Q is not closed and the complement of Q is neither (none of the points in Q are interior points so Q is not open and its complement R-Q has the same property so it is not open, therefore, Q is not closed)
E) from (B), R-Z is the complement of Z, so it is open.
 
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  • #2
here's what I've come up with for the accumulation points:
A=[0,2]
B=Z
C=empty set
D=sqrt(2)
E=empty set.

any help anyone?
 
  • #3
Your original post looks fine, but you might want to specify open/closed relative to the underlying metric space (which is probably R in each case). Also, Z closed in R isn't necessarily immediate by definition, but it's an easy thing to prove. Compactness shouldn't be hard to answer, especially if you know Heine-Borel.

Taking the standard definition of an accumulation point of a subset A of a metric space X as a point p in which every open ball about p which contains a point q in A, q =/= p, then I would take another look at your answers. I think you may be confusing the fact that a set is closed if it contains all of its accumulation points with the false notion that an open set has no accumulation points. Also, Z is closed relative to R, but the integers cannot be the accumulation points (by the standard definition). But if a set has no accumulation points, then it is closed, because the condition is vacuously satisfied.
 
  • #4
I, personally, would not use the word "neither" when there are three options.
 
  • #5
okay how about this:
A=not open, not closed and therefore, not compact; accum. pt: [0,2]
B=closed, and compact; accum pt: empty set
C=open, and not compact; accum pt: empty set
D=not open, not closed and therefore, not compact; accum pt: empty set
E=open, and not compact; accum pt: empty set

can you please check that?
 
  • #6
A and B look correct. However, why do you think that E for instance has no accumulation points? R-Z is just the reals with "holes" where in the integers are. If I pick a point at say 1.5, which is in R-Z, and look at any open interval about the point, I would find another real number different from 1.5 right?

*EDIT* Also, HallsofIvy raised a point I failed to catch the first time. Ignoring compactness, you should still state whether each set is open and whether each set is closed. A set could be open and closed, and the standard example is the whole space itself (trivially satisfies the definitions). Fortunately, I'm assuming we're working with R as the underlying space, and R has the property that the subsets of R that are simultaneously open and closed are exactly the whole set and the empty set. But this should just make it easier to answer to the question of both openness and closedness.
 
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  • #7
The integers are not compact.
 

1. What is an open set?

An open set is a subset of a topological space in which all points are interior points. This means that for every point in the set, there exists a small enough neighborhood around that point that is also contained within the set.

2. How is a closed set defined?

A closed set is the complement of an open set. This means that a set is closed if and only if its complement is open. In other words, a set is closed if and only if it contains all of its limit points.

3. What is the difference between an open and closed set?

The main difference between open and closed sets is that open sets only contain interior points, while closed sets contain both interior and boundary points. Another way to think about it is that open sets don't touch their boundaries, while closed sets include their boundaries.

4. What is a compact set?

A compact set is a set that is both closed and bounded. This means that the set contains all of its limit points, and it can be covered by a finite number of open sets. In other words, a compact set is a set that is not "too big" or "too spread out."

5. How do open, closed, and compact sets relate to each other?

Open and closed sets are complementary concepts, while compact sets are a combination of both. A set can be both open and closed (a clopen set) if and only if it is the entire space or the empty set. Compact sets are a special type of closed set, as they contain all of their limit points, but they are also not too "spread out" like open sets.

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