How to attempt the following question?
Question: Let $Y=[0,1]$, considered as a subspace of $\mathbb{R}$ with the topology $\tau=\{ I \subseteq \mathbb{R} \mid I\text{ an open interval}\}.$ Find the subspace topology.
Here are my steps:
- Let $I = (a,b)$. Let $\underline{T}={Y\cap U\mid U\tau T}$.
- $(a ,b) \cap Y= (a, b)$ if $a, b \in Y$.
- $(a, b)\cap Y=(0, b]$ if only $b \in Y$.
- $(a ,b)\cap Y= (a,1]$ if only $a \in Y$.
- $(a, b)\cap Y= Y$ or $\emptyset$ if neither $a$ nor $b$ are in $Y$. Then, $\underline{T}= \{ \emptyset, [0,b), (a,1], Y\}$.
I am not sure whether I can include $(a, b)$ in $\underline{T}$!
$\underline{T}=\{ \emptyset, [0,b), (a,1], Y\}$