This page is designed to help if you have a problem concerning open and/or closed sets, particularly in . Clicking on answers to the questions below will lead to suggestions or further questions.
Basic real analysis, the definitions of open and closed.
A piece of general advice
When thinking about open or closed sets, it is a good idea to bear in mind a few basic facts.
First, a subset of (or any metric space, but this does not apply to all topological spaces) is closed if and only if whenever is a sequence of elements of that converges to a limit , then that limit belongs to as well. In other words a set is closed (in the sense of having a complement that is open) if and only if it is closed under taking limits.
Second, if , then is continuous if and only if is an open subset of whenever is an open subset of . (Again, this holds for arbitrary metric spaces. It also holds for topological spaces, but then it is the definition of continuity.)
Third, a closed bounded subset of is compact (but a closed bounded subset of an arbitrary metric space does not have to be compact).
Fourth, a finite intersection of open sets is open and any union of open sets is open; and similarly a finite union of closed sets is closed and any intersection of closed sets is closed.
What is your problem?
Which of the following descriptions best fits your problem?
I am trying to prove that a certain set is open or closed.
- In that case, an obvious approach is to begin your proof by saying "Let ," and going on to try to prove that there must be some such that whenever the distance between and is less than . But often it is much cleaner to use the basic facts above. For some examples, see the article To prove that a set is open or closed, use basic theorems rather than direct arguments.
- I am trying to construct an open or closed set with a certain property.
- Sometimes you may be able to do this by defining your set to be , where is a set that you already know to be open or closed and is a continuous function. But in more complicated problems you may well need to express your set as a union of infinitely many simpler open sets or an intersection of infinitely many simpler closed sets. See the article To construct exotic sets, use limiting arguments for further details.
- I want to prove that an open or closed set has some other property.
- It is hard to give general advice about this situation, except that you should be alert to the possibility that a closed set is compact, which it will be, for example, if it is a closed bounded subset of or a closed subset of a compact metric space. If that is the case, then you have lots of facts about compactness to draw on. See How to use compactness for more about this.