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.

Clopen sets are both closed and open.  For example, the empty set and the entire space are always clopen.  Half-open intervals are not necessarily clopen; for example, $[0,1/2)$ is not closed, because it does not contain $1/2$.
One set that is clopen in the subspace topology is $[0,1]$.  Even though it is not open in $\mathbb R$,  it is open in the subspace topology, because it is the intersection of the subspace $[0,1]\cup[3,4]$ and $(-1,2), $ which is open in $\mathbb R$.
I have given you three clopen sets in the subspace topology, and I hope you can find a fourth.
