I wonder if it's possible for a topological space $(M, \tau)$ where $\tau$ is the collection of all open subsets of $M$, to have a finite Lebesgue covering dimension, while having no proper subspace of finite (but nonzero) dimension. Especially if such space is not Hausdorff and not locally Euclidean.
I am curious about the case in which the set $M$ doesn't have any isolated point.