I have a problem about open or closed sets

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I have a problem about open or closed sets

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[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 $f:\R^n\rightarrow\R^m$, 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?

*[add]I am trying to prove that a certain set $X$ is open or closed.{{
**In that case, an obvious approach is to begin your proof by saying "Let $x\in X$," and going on to try to prove that there must be some $\delta>0$ such that $y\in X$ whenever the distance between $x$ and $y$ is less than $\delta$. 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]].}}[/add]

*[add]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.}}[/add]

*[add]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.}}[/add]

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