general topology - Topological Embedding Which is Neither Open nor Closed

I'm having trouble coming up with an example of an embedding which is neither open nor closed.

My attempts have included trying to find such a map from $\mathbb{R}$ (given the usual Euclidean topology, of course) to some subset of $\mathbb{R}$, which I now believe impossible, and trying to find one from some topology on $\{1, 2, 3\}$ to some other topology on $\{1, 2, 3, 4\}$. Both of these attempts seem to have failed me. So what do I do?