Proper clopen subset of disconnected set

Question

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 $a<b<c<d$ should have a proper clopen subset. However, $\mathbb R$ has no clopen proper subsets and any proper subset of $U$ is also a proper subset of $\mathbb R$ which is a contradiction. Where am I wrong in my reasoning?

Thanks in advance!

Answer 1

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 $b<e<c$, then $[a,b]=U\cap(a-1,e)$. It is also closed in $U$ because $[a,b]$ closed in $\Bbb{R}$ and $[a,b]=U\cap[a,b]$.

Answer 2

$[a, b]$ 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 $(a-1, \frac{b+c}{2})\cap U$.