# In a metric space is a dense subset of a dense subspace dense in the space itself?

In a metric space is a dense subset of a dense subspace dense in the space itself?

I think that must be false, but I couldn't think of any examples of the contrary.

• Maybe you should the question more clearly in the body, because given how it's phrased right now it seems true to me. Jul 27, 2020 at 15:56

To add to the main answer. It's true for any topological space (no metric required). Suppose $$X$$ is a topological space, and $$A\subseteq B\subseteq X$$ with $$A$$ dense in $$B$$ and $$B$$ dense in $$X$$. Here: "$$A$$ dense in $$B$$" means $$A$$ is dense in the subspace topology on $$B$$.
Let $$U\subseteq X$$ be a nonempty open set. Then $$U\cap B$$ is nonempty since $$B$$ is dense in $$X$$. So $$U\cap B$$ is a nonempty open set in the induced topology on $$B$$. So $$U\cap B\cap A$$ is nonempty since $$A$$ is dense in $$B$$. So every nonempty open subset of $$X$$ intersects $$A$$ nontrivially, i.e., $$A$$ is dense in $$X$$.
It is true. Suppose $$(X,d)$$ is a metric space and we have inclusions $$A \subseteq B \subseteq X$$ where all the sets get the induced topology/metric. Suppose $$A$$ is dense in $$B$$ and $$B$$ is dense in $$X$$.
We show that $$\operatorname{cl}_X(A) = X$$, which will show denseness of $$A$$ in $$X$$.
Let $$x \in X$$. Let $$\epsilon > 0$$. Then since $$B$$ is dense in $$X$$ there is $$b \in B$$ with $$d(b,x) < \epsilon/2$$. Since $$A$$ is dense in $$B$$, there is $$a \in A$$ with $$d(a,b) < \epsilon/2$$. But then $$d(a,x) \leq d(a,b) + d(b,x) < \epsilon$$ so we have shown that $$x \in \operatorname{cl}_X(A)$$.