Is this true: Every open set $A$ contains a neighborhood whose closure is a subset of $A$. This seems a very easy fact. But I don't know how to prove it.
Can anybody help me? Thanks!
 A: Hint: there is a topology on the two-point set $\{a,b\}$ that will give you a counterexample.
A: This holds in a regular space ($T_3$ space) and above, including a normal space ($T_4$ space).

*

*A topological space $X$ is a regular space if, given any nonempty closed set $F$ and any point $x$ that $x\notin F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that $U\cap V=\varnothing$.


*A topological space $X$ is a normal space if, given closed sets $E$ and $F$ that $E\cap F=\varnothing$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that $U\cap V=\varnothing$.
Edit: Here is the proof of your claim.

Theorem: In a regular space ($T_3$ space), for any $x\in X$ and each open set $U$ that $x\in U$, there is an open set $V$ that $x\in V$, such that $\overline{V}\subset U$.

Proof: Let $X$ be a regular space and $x\in X$, $U$ be an open set that $x\in U$. So there exists another open set $O$ that $x\in O\subset U$. Since $O^c$ is closed and $x\notin O^c$, by 1, there exist open sets $O_1,O_2$ such that $x\in O_1, \:O^c\subset O_2$ and $O_1\cap O_2=\varnothing$.
Let $V=O_1$, then $V$ is open, and $x\in V\subset\overline{V}\subset O_2^c$ since $V\cap O_2=\varnothing$ and $O_2^c$ is closed. So $x\in V=\overline{V}\subset O\subset U$ since $O^c\subset O_2$ and $O\subset U$. $\tag{Q.E.D}$
Since a normal space is a regular space and $\Bbb{R^n}$ is normal, the above also holds for a normal space and $\Bbb{R^n}$. For $\Bbb{R^n}$, just replace any open set containing $x$ with a ball $B_r(x)$ centered at $x$.
A: If you are working in $\mathbb{R}^n$ then this is true; just take a very small ball contained in your open set.  
In a general topological space this is not true (assuming you want a nonempty set).  For example, in any irreducible space every nonempty open set is dense.  These kinds of spaces occur in algebraic geometry.  
