$X$ is a topological vector space. $K$ is convex subset of $X$. if $x\in int(K)$ and $y \in K$, then $$[x,y)=\{xt+(1-t)y| t\in (0,1]\} \in int(K)$$ .
for $t=1$ and if $K=\phi$, proof is trivial. I have done solution of this problem using normed $X$. How to proof this result in general?
I suppose your $K$ is convex (rather than being compact).
Let $u =int (K)$. Then $tU+(1-t)y $ is open and it is contained in $K$.Take union over $0<t\leq 1$ to see that $[x,y)$ is contained in an open set which is contained in $K$. Hence, $[x,y)$ is conatined in $int (K)$.
[For fixed $t >0$ and fixed $z \in X$ the map $x \to tx+z$ is a homeomorphism of $X$ by the defintion of a toplogical vector space. Hence, $tU+z$ is open whenever $U$ is open].