gn.general topology – Is the Cantor set zero-dimensional base?

Definition. A topological space of zero dimension. $ X $ is called zero-dimensional basis yes for any base $ mathcal B $ of the topology that consists of closed and open sets in $ X $, any lid open $ mathcal U $ of $ X $ has a disjointed refinement $ mathcal V subset mathcal B $.

It can be shown that each regular accounting space is a zero-dimensional basis.

Issue. It's the singer's cube $ {0,1 } ^ omega $ zero-dimensional basis?