Can an interval on the real line, $(a,b)\subset \mathbb R$ be connected if its inf and sup are not part of the interval? Obviously if the inf and sup of the interval are not in the interval it cannot be a closed interval.
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what about the interval $(a,b) \subset \mathbb{R}$? – Lionel Ricci Oct 18 '15 at 16:52
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Are there intervals that are not connected? (no) What is an "interval set" – just an interval? I almost don't understand the question. – BrianO Oct 18 '15 at 17:03
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Yes, I mean the set is simply just an interval – jessica Oct 18 '15 at 17:21
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If the sup and inf of an interval are not included in the interval. Is the interval still connected? – jessica Oct 18 '15 at 17:21
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Hi are you still there Brian? I want to know if the inf and sup of an interval are not part of the interval. Is the interval still connected? – jessica Oct 18 '15 at 17:40
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$(1,2)$ is connected
$\sup(1,2) \notin (1,2)$
$\inf(1,2) \notin (1,2)$
Further reading:
Intervals are connected and the only connected sets in $\mathbb{R}$
https://proofwiki.org/wiki/Subset_of_Real_Numbers_is_Interval_iff_Connected
