In Bert Mendelson's "Introduction to Topology" p. 159, i read the statement "A topological space $C$ can be a subspace of two distinct topological spaces $X$ and $Y$. In this event the relative topology of $C$ is the same whether we regard $C$ as a subspace of $X$ or $Y$."
If $J$ is the topology of $X$ and $I$ the topology of $Y$, then the relative topology of $C$ with respect to $X$ consists of sets of the form $C \cap O$, where $O \in J$. Similarly the relative topology with respect to $Y$ consists of sets $C \cap O'$ with $O' \in I$. I don't see why these two relative topologies are identical.