Then B is the basis for a half-open topology denoted by T" in R.
Show that (R,T") is first countable but not second countable.
How can I prove it ?
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Have you managed to prove it is first countable? Or have you managed to prove it is not second countable? – Asinomás Oct 16 '16 at 18:22
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Related, but different question:. OP reasked as requested with different question phrasing. – JMoravitz Oct 16 '16 at 18:24
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It's called the Sorgenfrey line.It also has another proper name. – DanielWainfleet Oct 16 '16 at 18:33
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For real $r$ consider $\{[r,r+q): r<q\in \mathbb Q\}.$
If $C$ is any base for $T''$ then for each real $r$ there exists $f(r)\in C$ such that $r\in f(r)\subset [r,r+1).$ Consider $\{f(r): r\in \mathbb R\}.$
DanielWainfleet
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