Closed set which does not have open subsets
Solution 1
Here is an example of such a set with infinitely many elements:
The set of integers in $\mathbb R$ is closed, and any nonempty subset of integers is closed as well.
Solution 2
In any Hausdorff space, any finite set is closed.
So in any connected Hausdorff space a singleton does contain any nonempty open set.
(I mentioned connectedness as then the singleton cannot be open since it is already closed.)
Solution 3
The empty set is closed and does not contain any nonempty open set. Probably you meant to ask for a nonempty closed set which does not contain any nonempty open set. The following theorem answers this question in a general topological space X.
Theorem. If in the space $X$ there is a closed set which is not open, then there is a nonempty closed set which does not contain any nonempty open set.
Proof. If there is a closed set which is not open, then its complement, call it $U,$ is an open set which is not closed. Of course $U\ne\emptyset,$ since $\emptyset$ is closed. Assuming the axiom of choice, we can extend $\{U\}$ to a maximal collection $\mathcal U$ of pairwise disjoint nonempty open sets. Then the set $A=X\setminus\bigcup\mathcal U$ is a closed set which does not contain any nonempty open set. I claim that $A\ne\emptyset.$
If we had $A=\emptyset,$ it would follow that $\bigcup\mathcal U=X$ and that $\bigcup(\mathcal U\setminus\{U\})=X\setminus U,$ so that $X\setminus U$ would be an open set, contradicting the fact that $U$ is not closed.
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wlaudsla123
Updated on July 27, 2020Comments

wlaudsla123 over 3 years
I wonder if there is an example of a set $A$ that is closed and that does not contain any nonempty open set.

M. Winter over 6 yearsTake a single point for example.

wlaudsla123 over 6 yearssince single point have no limit point , it is closed set??

M. Winter over 6 yearsA single point is its own and only limit point. A set with a single point does only have the constant sequence which converges to the only point present. So it contains all its limit points, hence is closed.

Riccardo Ceccon over 6 yearsYou should specify if you're talking about $A$ as a subset of $\mathbb{R}^n,$ a metric space, a topological space etc...

wlaudsla123 over 6 yearsA as a subset of a metric space

Martin Sleziak over 6 yearsI will just point out that not containing a nonempty open set is equivalent to having empty interior. Such sets are also called codense (see, for example, this answer) and less frequently boundary set or border set (probably influenced by Polish authors  it's called Zbiór brzegowy in Polish).
