Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space PLJ of Lorentzian polynomials on J modulo R>0
Nearby in the stack
, which is nonempty if and only if
J
is the set of bases of a polymatroid. We prove that
PLJ
is a manifold with boundary of dimension equal to the Tutte rank of
J
, and more precisely, that it is homeomorphic to a closed Euclidean ball with the Dressian of
J
removed from its boundary. Furthermore, we show that
PLJ
is homeomorphic to the thin Schubert cell
GrJ(Tq)
of
J
over the triangular hyperfield
Tq
, introduced by Viro in the context of tropical geometry and Maslov dequantization, for any
q>0
. This identification enables us to apply the representation theory of polymatroids developed in a companion paper, as well as earlier work by the first and fourth authors on foundations of matroids, to give a simple explicit description of
PLJ
up to homeomorphism in several key cases. Our results show that
PLJ
always admits a compactification homeomorphic to a closed Euclidean ball. They can also be used to answer a question of Brändén in the negative by showing that the closure of
PLJ
within the space of all polynomials modulo
R>0
is not homeomorphic to a closed Euclidean ball in general. In addition, we introduce the Hausdorff compactification of the space of rescaling classes of Lorentzian polynomials and show that the Chow quotient of a complex Grassmannian maps naturally to this compactification. This provides a geometric framework that connects the asymptotic structure of the space of Lorentzian polynomials with classical constructions in algebraic geometry.