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I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact?

I'm not sure if a countable union of countable unions is still a countable union.

Thanks.

Asaf Karagila
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Talexius
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    http://www.proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable –  Jan 21 '14 at 19:02
  • I seem to recall that a year or so ago someone asked here how to prove that a union of countably many countable sets is countable and I, and several others, answered. – Michael Hardy Jan 21 '14 at 19:12
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    Here and here. – Michael Greinecker Jan 21 '14 at 19:25
  • @MichaelGreinecker and T.Bongers, Guys, please read again. I'm not asking if a countable union of countable sets is countable. If what I wrote of countable unions is not clear, the real answer is above. I'm asking if $X$ is a countable union of $\sigma$-compact sets then $X$ is $\sigma$-compact. I wrote nothing about countable sets. – Talexius Jan 21 '14 at 19:40
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    @Talexius Let $I$ be some index set and for each $n$, $C_n\subseteq I$ be countable. Then $\bigcup_n \Big(\bigcup_{i\in C_n} U_i\Big)=\bigcup_{i\in\bigcup_n C_n}U_i$. – Michael Greinecker Jan 21 '14 at 19:48
  • Thank you @MichaelGreinecker ! If you post your reply I could select it as the answer. – Talexius Jan 21 '14 at 20:43
  • @Talexius I think it is more productive if you post it as answer. – Michael Greinecker Jan 21 '14 at 21:12

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