Open subsets of the closed unit interval.

Question

Can anyone help me in solving this problem:

I feel like I should use the definition of a subspace topology, but I do not know how.

Answer 1

Remember that a subset $U \subset I$ is open in $I$ if it is the intersection of $I$ with an open set from $\mathbb{R}$. So write

$U = I \cap W$

where $W \subset \mathbb{R}$ is open. $0 \in U$, so $0 \in W$, and since $W$ is open, it will contain an open interval around $0$, say, $(-a, a)$, $a>0$. Therefore, $U = I \cap W$ will contain the interval $[0,a)$.

Now, let $W^\prime = W \cap I$. This is open in $I$ and contains $[0,a)$ (if we assume $a \le 1$), so we have

$U = W \cap I = W^\prime \cap I = W^\prime = [0,a) \cup W^\prime$

The last equation feels a bit odd, but the important thing to show is that $U$ contains the interval $[0,a)$.