TITLE: Not open and not closed set
QUESTION [5 upvotes]: Can somebody provide not open and not closed set? I even cannot imagine what does it mean. Also, I'm having a deal with such problem that there is a bounded countable set in R, and I should provide examples of sets that are: opened; closed; not opened and not closed; compact, which means bounded and closed.

REPLY [3 votes]: Hint:
A bounded countable set in $\mathbb R$ is easy to find if you know that $\mathbb Q$ is a countable set. Can you find a bounded subset of $\mathbb Q$?
All of the other examples can be intervals in $\mathbb R$. Just play around with whether the edge points are included or not.

REPLY [2 votes]: The interval $[0,1)$ is neither open nor closed in the usual topology on $\mathbb{R}$.
A more interesting example of a subset of $\mathbb{R}$ which is neither open nor closed is the set of irrational numbers. It is not open since it is not the union of open intervals (any open interval $(a,b)$ contains a rational number), and it is not closed, since it does not contain all its limit points (there is a sequence of irrational numbers which limits to $0$, e.g. $\pi$, $\pi/2$, $\pi/3$, $\dots$, $\pi/n$,$\dots$, but $0$ is not irrational).