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I cannot seem to even be able to get started on this Topology question.

What we know: A $\subset$ X is dense in itself => A $\subset$ A'.

What we want: U $\subset$ A, U open such that U $\subset$ U'.

Am I at least on the right track with writing the above statements down?

Thank you!

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    Your title makes no sense to me. Does it say what you want it to say? – 2017-02-01

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A space is dense in itself if it doesn't contain isolated points. In other words $X$ is dense if none of the singletons $\{x\}$ are open.

Now if $X$ is a space and $U\subseteq V\subseteq X$ are such that $V$ is open in $X$ and $U$ is open in $V$ then $U$ is open in $X$.

In particular if $\{x\}$ were open in any open subset $U\subseteq X$ then it would be open in whole $X$.