In my topology textbook (Bert Mendelson's) it is stated that if $C$ is a subspace of two distinct larger spaces $X$ and $Y$, then the relative topology of $C$ is the same whether we regard it as a subspace of $X$ or $Y$.
In an attempt to prove this, regard $C$ as a subspace of $X$ and suppose $S$ is an open subset of $C$. Then $$S = S' \cap C$$ for some open subset $S'$ of $X$. Now I'm not sure how to continue; proving that $S' \cap Y$ is an open subset of $Y$ would do the job, since then $$S' \cap Y \cap C = S$$ is in the topology of $C$ regarded as a subspace of $Y$, but I don't see a way to prove this.
Can anyone help me with this proof? Can $S' \cap Y$ be shown to be open in $Y$ or should I take a completely different approach?
Thanks in advance!