on the interior of a subset [closed]

Let $X$ be a topological space and let $K$ be a subset of $X$. Suppose that there is a closed set of $X$, say $C$, such that $K\subseteq C$. So, is it true that saying that $K$ has non empty interior in $C$ is the same thing to say that $K$ has non empty interior in $X$ ? thank you

closed as off-topic by user26857, Leucippus, Daniel W. Farlow, JonMark Perry, Ben ShellerMar 23 '16 at 1:50

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• It is true if $K\subset U$, $U$ open – Tsemo Aristide Feb 17 '16 at 0:42

No, consider the $x$-axis in $\mathbb{R}^2$; a subset of it can be open in the relative topology (think to an open interval) so be equal to its interior (in the relative topology), but it has empty interior as a subset of $\mathbb{R}^2$.
No. Consider the case where $K$ is a non-empty closed subset of $X,$ with empty interior in $X,$ and $C=K.$ Then Int$_CK=$Int$_KK=K\ne \phi$ but Int$_XK=\phi.$ Example: In the reals, let $K=C=\{0\}.$