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Let $X$ be a topological space (i) We call $X$ connected if there does not exist a pair of disjoint nonempty open sets whose union is $X$. (ii) We call $X$disconnected if $X$is not connected.(iii) If $X$is disconnected, then a pair of disjoint nonempty open sets whose union is $X$is called a separation of $X$.

Then, $B= \{[a,b)\ s.t.\ a,b \in \mathbb{R}\ and \ a < b\}$ (lower-limit topology) results in a disconnected topological space (in fact it's totally disconnected). The same could be said about $B= \{(a,b]\ s.t.\ a,b \in \mathbb{R}\ and \ a < b\}$, the upper-limit topology. And of course the discrete topology results in a disconnected topological space.

Are my understandings correct?

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  • $\begingroup$ The first two are not in fact topologies. $\endgroup$ – Tobias Kildetoft Dec 21 '16 at 12:51
  • $\begingroup$ We can define the upper/lower limit topology on $\mathbb{R}$ via those basis, right? $\endgroup$ – Learner Dec 21 '16 at 12:54
  • $\begingroup$ Yes, B is a base for a topology. The lower-limit topology is also known as the Sorgenfrey line. $\endgroup$ – DanielWainfleet Dec 21 '16 at 18:08
  • $\begingroup$ OK - Do the upper and lower limit topologies result in a disconnected topological space like I mentioned? $\endgroup$ – Learner Dec 21 '16 at 19:26
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    $\begingroup$ @Learner: Yes, they do. $\endgroup$ – Brian M. Scott Dec 21 '16 at 20:14

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