Find all compact sets in $\mathbb{R}$

Question

In $\mathbb{R}$, considering the topology consisting of the empty set and all sets containing $0$ and $1$, I need to find all compact sets.

I understand the definition of a compact set but don't know how to apply it to this situation.

I also need to find all the compact sets of a topology consisting of $\mathbb{R}$ and all sets NOT containing $0$ and $1$.

Answer 1

Hint: $\{0,1,2\}$ is open. The topology looks a lot like the discrete topology on $\Bbb R \setminus \{0,1\}$

Answer 2

For the first topology :- The only compact sets are finite sets. Clearly finite sets are compact so we just need to prove that any infinite set is not compact. Let $A$ be any infinite set. Consider the open cover $\{0,1,a\}_{a\in A}$. Then since $A$ is infinite this open cover does not admit any finite subcover and hence $A$ is not compact.

Hint for the second topology :- Notice that in this topology $\{a\}\mid a\neq 0,1$ is an open set. So can you identify all the compact sets now ?

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Edit :- (Elaborated Hint) For the second case again let $A$ be any infinite set. Is $\{a\mid a\in A\}$ an open set ? If yes, then can you find an open cover for $A$ which does not have any finite subcover ?