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Somebody can to give me a simple example of Lindelöf space?

Note. Lindelöf space is a topological space in which every open cover has a countable subcover.

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The reals are Lindelof. In fact any second countable space is Lindelof. – Forever Mozart Jun 27 '13 at 22:13
I guess you want an example in which the subcovers are countable but not finite, do you? You should put that in the question. – MyUserIsThis Jun 27 '13 at 22:14
up vote 3 down vote accepted

The natural numbers with the discrete topology.

Given an open cover, $U_i$ let $U_n$ be some open set such that $n\in U_n$, then $\{U_n\mid n\in\Bbb N\}$ is a countable subcover.

Although simpler example, perhaps, would be any compact set. I still think that you may benefit from a non-compact example.

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