Valid method to obtain a basis of a topological subspace? Let $(X,\tau)$ be a topological space and $Y \subset X.$ We know that if $\mathcal{B}$ is a basis for $\tau$ and $\tau_{\small{Y}}$ is the subspace topology on $Y$, then we can obtain a basis for $\tau_{\small{Y}}$
by taking the collection $\mathcal{B}_Y$ of intersections $Y \bigcap B$ as $B$ ranges over all the sets in $\mathcal{B}$.
I was wondering if as another genereal method to obtain a basis for $\tau_{\small{Y}}$ by taking the collection $\mathcal{S}_Y$ of all the sets $B$ in $\mathcal{B}$ such that $B \subset Y$? If yes, will we always have that $\mathcal{B}_Y=\mathcal{S}_Y$?
As an example I took the interval $[1,2]$ with the induced Euclidean topology and it seems to work.

No. Take a line in $\mathbb{R}^2$.
Also, your case $[1,2]$ doesn't seem to work for me. It seems neither $1$ nor $2$ would have a neighbourhood.
