Non-compactness of $\mathbb{R}$ with the cocountable topology

Question

Is $(\mathbb{R},\tau_{co})$ compact where $\tau_{co}$ is the cocountable topology on $\mathbb{R}$?

I have the answer of my teacher but I'd like to see another one so I can understand better how people find the answer intuitively. He says:

$\mathbb{N}-\{n\}$ is infinite countable for all $n\in\mathbb{N}$. Then $V_n = \mathbb{R}-(\mathbb{N}-\{n\})=(\mathbb{R}-\mathbb{N})\cup\{n\}\in\tau_{co}$ for all $n\in\mathbb{N}$.

Then $U=\{ V_n : n\in\mathbb{N}\}$ is a open covering of $(\mathbb{R},\tau_{co})$.

$U$ does not contain a countable subcover since $V_n\cap\mathbb{N}=\{n\}$ for each $n\in\mathbb{N}$ and $\mathbb{N}$ is infinite.

How would you solve that problem?

Answer 1

Another way is to make use of the characterisation of compactness by families of closed sets with the finite intersection property:

A topological space $X$ is compact iff whenever $\mathcal{A}$ is a family of closed subsets of $X$ with the finite intersection property (i.e., the intersection of any nonempty finite subfamily of $\mathcal{A}$ is nonempty), then $\bigcap \mathcal{A} \neq \varnothing$.

Note that the closed subsets of $( \mathbb R , \tau_{\text{co}} )$ are exactly the countable subsets of $\mathbb{R}$ (and $\mathbb R$ itself). It follows that for each $n \in \mathbb{N}$ the set $A_n = \{ k \in \mathbb N : k \geq n \}$ is a closed subset of $( \mathbb R , \tau_{\text{co}} )$. It is easy to see that $\{ A_n : n \in \mathbb N \}$ has the finite intersetion property (if $n_1 < n_2 < \ldots < n_\ell$ are natural numbers, then $A_{n_1} \cap A_{n_1} \cap \cdots \cap A_{n_\ell} = A_{n_\ell}$ is nonempty). However $\bigcap_{n \in \mathbb{N}} A_n = \varnothing$.

This can be translated into essentially your instructor's proof, but in co-finite and co-countable topologies I find that it is often easier to work with the closed sets as opposed to the open sets.

Answer 2

For $q \in \mathbb Q$, let $\mathcal O_q = (\mathbb R \setminus \mathbb Q) \cup \{q\}$ i.e. the set of all irrational numbers and $q$. Then, $\mathcal O_q$ is cocountable and $\mathbb R = \bigcup_{q\in \mathbb Q} \mathcal O_q$ but no finite subcover would cover all of the rational numbers.