  We associate to any compact semi-algebraic set $X \subset \mathbb R^n$ a
chain complex of currents $S_\ast (X)$ generated by integration along
semi-algebraic submanifolds and we analyze the corresponding homology groups.
In particular, we show that these homology groups satisfy the
Eilenberg-Steenrod axioms and further, that they are isomorphic to both the
ordinary singular homology groups of $X$ and to the homology groups generated
by the integral currents supported on $X$. Using this result and a certain
neighborhood of $X$, we are able to prove homological mass minimization for
integral currents supported on $X$, and verify that any cycle of $X$ that has
sufficiently small mass is a boundary.
