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Consider the space $(\mathbb{R}, \mathcal{T}_{\text{coen}})$ from where $\mathcal{T}_{\text{coen}}$ is the topology of countable complements (co-countable topology), i want to show that $(\mathbb{R}, \mathcal{T}_{\text{coen}})$ is a Lindelof space.

The idea would be to consider an open cover of $\mathbb{R}$, namely $\mathcal{C}$ if $A\in \mathcal{C}$ let

$$A=\mathbb{R}-\{x_{i} \}_{i=1}^{\infty}=A_{0} \hspace{0.3cm} \text{i think} \hspace{0.3cm} x_{i}\in \mathbb{R}$$

Let $A_{1}\in \mathcal{C}$ such that $x_{1}\in A_{1}$ and $A_{2}\in \mathcal{C}$ such that $x_{2}\in A_{2}$, in general $A_{k} \in \mathcal{C}$ s.t. $x_{k}\in A_{k}$, note that

$$\mathcal{C}'=\{A_{1}, \cdots, A_{k} \cdots \}$$

Is an open cover countable of $\mathbb{R}$ s.t.

$\mathcal{C}' \subseteq \mathcal{C}$

Maybe this can work, but I'm certainly not sure if this argument can be improved, any suggestions? thanks.

Wrloord
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1 Answers1

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Your idea is right. Take some open cover $\mathcal{C}$ of $X$ (where $X$ is $\mathbb{R}$ with co-countable topology).

Pick some $A\in \mathcal{C}$ first, then for every $x\in X\setminus A$ pick some set $A_x$ with $x\in A_x\in\mathcal{C}$.

Then $\mathcal{C}' = \{A\}\cup \{A_x : x\in X\setminus A\}$ is a countable subcover of $\mathcal{C}$, so $X$ is Lindelof.

Jakobian
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