Past exam of the topology course of the University of Oxford / 2021 / 1 / b / iv / Solution

The only isolated point of the subset $[0,1]$ is $1$. Indeed,so $\{1\}$ is open in the subspace topology. If $0\le x<1$, every basic neighbourhood $[a,b)$ containing $x$ also contains a point strictly between $x$ and $\min (b,1)$, hence another point of $[0,1]$. Such an $x$ is not an isolated point of the subset. Here isolation refers to $[0,1]$ with its subspace topology; $1$ is not an isolated point of the ambient Sorgenfrey line.