  An intersection of sets $A = \bigcap_{i \in I}B_i$ is irredundant if no $B_i$
can be omitted from this intersection. We develop a topological approach to
irredundance by introducing a notion of a spectral representation, a spectral
space whose members are sets that intersect to a given set $A$ and whose
topology encodes set membership. We define a notion of a minimal representation
and show that for such representations, irredundance is a topological property.
We apply this approach to intersections of valuation rings and ideals. In the
former case we focus on Krull-like domains and Pr\"ufer $v$-multiplication
domains, and in the latter on irreducible ideals in arithmetical rings. Some of
our main applications are to those rings or ideals that can be represented with
a Noetherian subspace of a spectral representation.
