How many clopen subsets does the subspace [0,1]∪[3,4] have? Show that the subspace $[0,1]∪[3,4]$ of $\mathbb R$ has at least $4$ clopen subsets. Exactly how many clopen subsets does it have?
Just to clarify, this is the subspace induced by $[0,1]∪[3,4]$ from $\Bbb R$, so the space is just the intersection of all open sets in $\Bbb R$ with $[0,1]∪[3,4].$
Finding $4$ clopen sets is easy. Just form half open intervals with one of the $4$ endpoints, like $[0, 1/2).$ As for how many clopen sets there are, I think the answer is that there are  uncountable many, since the intervals $(0,1)$ and $(3,4)$ have uncountably many elements.

$[0,1]$ and $[3,4]$ are connected spaces. So they have no non-trivial clopen subsets. So if $C$ is clopen in $X=[0,1]\cup[3,4]$, then $C \cap [0,1]$ is also clopen in $[0,1]$ and so can only be $\emptyset$ or $[0,1]$, and, likewise, $C \cap [3,4]$ is $\emptyset$ or $[3,4]$. These $2 \times 2$ options give you four options for $C$, and it's easy to check that these are all clopen.
