Connected Properties of an Interval Set 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.

$(1,2)$ 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
Set $A$ interval in $\mathbb{R}\implies$ connected
