Let $(X,\tau)$ be a topological space.


We say that $(X,\tau)$ is *zero-dimensional with respect to the Lebesgue covering dimension (zd1)* if every open cover of the space has a refinement which is a cover of the space by open sets such that any point in the space is contained in exactly one open set of this refinement.


Moreover, $(X,\tau)$ is *zero-dimensional with respect to the small inductive dimension (zd2)* if it has a base consisting of clopen sets.


Is there a space that is (zd1) but not (zd2)?


**EDIT**: I accepted the small and correct example given by Gabriel below; it works for the reason that $X\in \mathcal{V}$ for every open cover $\cal V$. It would be great to see an example of a space $(X,\tau)$ that is (zd1) but not (zd2), and such that $X$ has a cover $\cal V$ such that $X\notin \mathcal{V}$.



Two notions of zero-dimensionality for topological spaces
Take the Sierpiński space $X=\{0,1\}$ with $\tau=\{\{\},\{0\},\{0,1\}\}$.


