Why can't we compute the singular cohomology of schemes/varieties with the Zariski topology?

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I have been reading etale cohomology. The book says that it is algebraic analogue of singular cohomology.

My question is that why can’t we compute the singular cohomology of schemes/varieties over the Zariski topology?

JS Milne states many reasons for the inadequacy of Zariski topology in computing sheaf cohomology. But I want to know the short comings of the Zariski topology in calculating singular cohomology (as in algebraic topology).