$\mathbb R_l$ is not connected.
3,594
Solution 1
Any open set $[a,b)$ can be written as $[a,c) \cup [c,b)$ for $c$ between $a$ and $b$, so all open sets can be decomposed into non-empty, disjoint open sets.
Solution 2
The sets $[0,\infty)$ and $(-\infty,0)$ are both open in $\Bbb R_l$.
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Gauss
Updated on August 01, 2022Comments
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Gauss over 1 year
How to show $\mathbb R_l$ (lower limit topology on $\mathbb R$) is not connected?Means how any basis element of $\mathbb R_l$ can be written as the union of two separated sets?
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Angina Seng over 6 yearsCan you please remind us what $\Bbb R_l$ is?
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Admin over 6 years@LordSharktheUnknown $\mathbb R_l$ is a well-known notation for $\mathbb R$ with the lower-limit topology (the topology generated by the basis $\{[a,b): a,b \in \mathbb R\}$).
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Angina Seng over 6 years@OpenBall Could you then please remind us of a basis of $\Bbb R_l$?
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Admin over 6 years@LordSharktheUnknown sure! I just edited my comment.
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Gauss over 6 years:i just read $\mathbb R_l$ is finer than $\mathbb R$ but not getting how to use this fact in the problem.thanks a lot!!