Bott & Tu in *Differential forms in Algebraic Topology* write in Remark 5.17, pg.52

The two Poincare duals of a compact orientated submanifold correspond to two homology theories - closed and compact homology. Closed homology has now fallen into disuse, while compact homology is known these days as the homology of singular chains...in general Poincare duality sets up an ismorphism between closed homology and de Rham cohomology, and between compact homology and compact de Rham cohomology.

What is closed homology, and why did it fall into disuse? Did a better alternative turn up?

Differential Forms in Algebraic Topology. $\endgroup$ – AaronS Jul 8 '18 at 13:20