  This paper extends Alexander duality to the setting of parametrized homology.
Let X with be a compact subset of R^n x R (n \geq 2) satisfying certain
conditions, let Y be its complement, and let p be the projection onto the
second factor. Both X and Y are parametrized spaces with respect to the
projection. The parametrized homology is a variant of zigzag persistent
homology that measures how the homology of the level sets of the space changes
as we vary the parameter. We show that if (X, p|_X) has a well-defined
parametrized homology, then the pair (Y, p|_Y) has a well-defined reduced
parametrized homology. We also establish a relationship between the
parametrized homology of (X, p|_X) and the reduced parametrized homology of (Y,
p|_Y).
