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I have a problem about open or closed sets

Quick description

This page is designed to help if you have a problem concerning open and/or closed sets, particularly in \R^n. Clicking on answers to the questions below will lead to suggestions or further questions.

Prerequisites

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 X of \R^n (or any metric space, but this does not apply to all topological spaces) is closed if and only if whenever (x_n) is a sequence of elements of X that converges to a limit x, then that limit x belongs to X 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 \R^n\rightarrow\R^m, then f is continuous if and only if f^{-1}(U) is an open subset of \R^n whenever U is an open subset of \R^m. (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 \R^n 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 construct an open or closed set with a certain property.
    • Sometimes you may be able to do this by defining your set X to be f^{-1}(Y), where Y is a set that you already know to be open or closed and f 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 \R^n 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.