Are $[0, 1)$ and $[0, 1]$ homeomorphic subspaces of the Sorgenfrey line?

Question

My argument is that $\{1\}$ is a connected clopen subspace of $[0, 1]$ while the only connected subspaces of $[0, 1)$ are singular sets, which are not open in $[0, 1)$, so the spaces must not be homeomorphic. Is this reasoning correct?

Is there another argument that does not need require to test connectedness?

Thank you!

Answer 1

Your argument is correct, but you can just say that $[0,1]$ has a clopen singleton (which is $\{1\}$), whereas $[0,1)$ has none.