Relations between closed and compact sets

Question

I have the following doubts: consider the set $\mathcal{Y}=[0,1]$ which is closed, convex, compact. Let $\mathcal{F}$ be the collection of closed subsets of $\mathcal{Y}$ and $\mathcal{K}$ be the collection of compact subsets of $\mathcal{Y}$.

1) Can we say that $\mathcal{F}=\mathcal{K}$?

2) Let $K \subset \mathcal{K}$ and $K \neq \mathcal{Y}$. Can we say that it cannot be $\mathcal{Y} \subset K$?

Answer 1

1) It is a theorem that for a compact topological space, every closed subset is itself compact. See any textbook on topology.

2) This doesn't make sense. If $K$ is a compact subset of $\mathcal Y$, then clearly $K \subset \mathcal Y$, by definition of subset!