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In Chapman Pugh's Real Analysis the definition of a disconnected set is that it has a proper clopen subset. I was trying to apply that to a simple example but got the following inconsistency:

The disconnected set $U=[a,b]\cup[c,d]\subset\mathbb R$ where $a

Thanks in advance!

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    Here is the thing: your subset has to be clopen _in $U$_. In general topology, there is no "absolute" notion of open/closedness, and you always talk about a set being clopen _in some space_. In the definition of disconnectedness, this space is $U$. – 2017-01-15
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    Does the book say "nonempty proper clopen subset?" Because the empty set is also clopen. – 2017-01-15
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    @MattSamuel actually in the textbook he defines a proper subset as being nonempty. – 2017-01-15
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    That's pretty bizarre to me, but it does resolve the problem. – 2017-01-15
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    Rather similar older question: [Let $X = [0,1] \cup [2,3]$ be a metric space. Why is $[0,1]$ both open and closed?](http://math.stackexchange.com/q/1003063) – 2017-01-16

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You're mistaking the idea of being "open in $U$" with "being open in $\Bbb{R}$".

Indeed, $[a,b]$ is open in $U$ with the subspace topology because if you choose any $e$ with $b

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    Indeed, I think one of the biggest misconceptions people have with basic topology is the idea that a notion like "open" is a property of a set, when instead it's a relationship between a set and a space. – 2017-01-15
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$[a, b]$ is clopen in $U$, by the definition of subspace topology. It is closed because it's equal to $[a, b]\cap U$, and it's open because it's equal to $(a-1, \frac{b+c}{2})\cap U$.