Is there an example of family of open intervals in $\mathbb R$ such that any arbitrary union of such open intervals is again an open interval? In other words can we define a topology on $\mathbb R$ with open intervals only?, (i.e. $A$ is open in $\mathbb R$ if and only if $A$ is an open interval) However we have such a topology when we take such collection as base.
Is there an example of family of open intervals in $\mathbb R$ ......?
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general-topology
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1For a family of such open intervals, just make them nested. – 2012-06-30
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0@Kamran: Yes, it is an open interval, namely the open interval $(-\infty,\infty)$. It's open, and it is an interval because given any two $a,b$ in the set, $[a,b]$ is contained in the set. That's the definition of an interval. – 2012-06-30