  Jordan operator algebras are norm-closed spaces of operators on a Hilbert
space with $a^2 \in A$ for all $a \in A$. We study noncommutative topology,
noncommutative peak sets and peak interpolation, and hereditary subalgebras of
Jordan operator algebras. We show that Jordan operator algebras present perhaps
the most general setting for a `full' noncommutative topology in the
C*-algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for
not necessarily selfadjoint algebras by the authors with Read, Hay and other
coauthors. Our breakthrough relies in part on establishing several strong
variants of C*-algebraic results of Brown relating to hereditary subalgebras,
proximinality, deeper facts about $L+L^*$ for a left ideal $L$ in a C*-algebra,
noncommutative Urysohn lemmas, etc. We also prove several other approximation
results in $C^*$-algebras and various subspaces of $C^*$-algebras, related to
open and closed projections, and technical $C^*$-algebraic results of Brown.
