- #1
Ed Quanta
- 297
- 0
Which intervals [a,b] in the set of reals have the property that (the set of rationals and [a,b]) is both open and closed in the set of rationals? And the property closed but not open?
This doesn't prove that it's open, but it's on the right track. You have to show something similar for being able to find a b < a, which you can easily do. But I would also try to be more precise. The above would be enough to show that it is open, but that's because it implies something else, which in turn, by definition, implies that the set is open. To be rigorous, you should show these implications (and it should be very easy).This set should be open in Q since for any a in Set S, we can find b>a such that b is also in S
But 3^1/2 is not in S. Note that S only contains rationals, and the root of 3 is irrational. This set is not open in the reals, but for a different reason. If S is open in R, then for each x in S, there is an open interval I such that x is in I, and I is a subset of S. Note that an open interval in Q is different from an open interval in R. An open interval of Q around each x in S exists such that Q is a subset of S. An open interval of R around some x in S would contain rationals and irrationals, but since S contains only rationals, this open interval cannot be a subset of S.This set is not open in the Reals since when a= 3^1/2, we cannot find b>a such that b is in S
Ed Quanta said:Good point. I forgot that 3^1/2 wouldn't be included in my set since it is irrational. How do I write down the complements for set S then? Can I still write what I previously wrote down?
inquire4more said:I get the impression he's trying to approach it from a purely set theoretic treatment, such as for a class, and may not have the benefit of treating it topologically.
Closed and open sets refer to subsets of a larger set in mathematics. A closed set is a set that includes all of its boundary points, while an open set does not include its boundary points.
The main difference between a closed and open set is their boundary points. A closed set includes all of its boundary points, while an open set does not. Another way to think about it is that a closed set is "sealed" and contains all of its elements, while an open set is "open" and may or may not contain all of its elements.
To determine if a set is closed or open, you can look at its boundary points. If all of the boundary points are included in the set, then it is a closed set. If none of the boundary points are included, then it is an open set.
Closed and open sets are important in mathematics because they help us define and understand boundary points, which are crucial for concepts like continuity and convergence. They also have many applications in areas such as topology and analysis.
Yes, a set can be both closed and open. This type of set is known as a clopen set. In some cases, a set may be neither closed nor open, which means it does not include all of its boundary points and does not have an open boundary. This type of set is known as a non-closed and non-open set.