Is it true that subset D is dense in space X iff every nonempty open set in X contains one point of D I have a proposition that I am not sure if it is right: 

Let D be a subset of space X. D is dense in X if and only if every nonempty open set U of X contains at least one point of D.

When X is a metric space, then I could verify that this is true in both directions.

When X is a topological space, I could prove the ==> direction; however the &lt;== direction is so difficult that I doubt the proposition is right. However, I cannot yet find a counter example.

This proposition is the generalization of a property of real numbers R:

Q is dense in R so every open interval of R contains a point of Q.

Thank you.
As to your definition: $D$ is dense when the closure of $D$ (a.k.a. $\overline{D}$) in $X$, equals $X$. (this does not mean in general that every point of $X$ is a limit of a convergent sequence from $D$; it does in metric spaces, but not in general topological spaces). So as the definition I will use $\overline{D} = X$.

Of course, now the solution will depend on the definition of the closure of a set $A$. Let's use the definition, or characterization, that $x \in \overline{A}$ iff for every open set $O$ of $X$ that contains $x$, $O$ intersects $A$.

Now the statement about the density of $D$ is pretty clear: if $D$ is dense and $O \neq \emptyset$ is open, then pick $p \in O$, then $p \in \overline{D}$, as $\overline{D} = X$, and so every open neighbourhood of $p$ intersects $D$; in particular $O$ intersects $D$, i.e. $O$ contains a point of $D$.

On the other hand, if every non-empty open set $O$ of $X$ contains a point of $D$, let $x \in X$. Let $O$ be any open set that contains $x$. Then $O$ is non-empty (because of $x$) and so contains a point of $D$, i.e. $O \cap D \neq \emptyset$. But this just says that $x \in \overline{D}$ by the definition above. As $x \in X$ was arbitrary, $X \subset \overline{D}$. As $\overline{D} \subset X$ by definition, we have equality.

Note that nothing at all of a metric is used. This is just a general fact for topological spaces.
I think your question is same with this question " A set D in a space X is dense in X iff every non-empty open set in X meets D", right?