Usually it works the other way around: things appear in topology first and then people realize that those things may have analogs in algebraic geometry. Etale cohomology is perhaps the best known example.

But let me give a counter-example (I wrote something similar as a comment to a recent question but I can't find it now). In topology there is the Lefschetz formula, which expresses the alternating sum of the traces of the cohomology endomorphisms induced by a smooth self-mapping of a smooth manifold in terms of the local contributions of the fixed points, assuming those are non-degenerate. There is a generalization of this for self-maps of arbitrary finite CW-complexes. The contribution of each fixed point is local, i.e. it can be determined by looking at the map in an arbitrarily small neighborhood of the point. In particular, if there are no fixed points, the alternating sum of the traces is zero.

Inspired by this, Grothendieck proved in SGA 5 an algebraic version of the Lefschetz formula, without smoothness or completeness assumptions. It also works for more general sheaves than the constant sheaf. Inspired by this, Goresky and MacPherson gave a topological version of the formula, which, under some assumptions, allows one to calculate the contribution of each component of the fixed point set. See "The local contribution to the Lefschetz fixed point Formula", Inv. Math. 111, 1993, 1-33.