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### Abstract

In this talk, we study positive inductive definitions over a class of T_{0}
quotients of countably based spaces (qcb_{0} spaces).
In particular, we prove that if A and B are qcb_{0} spaces, then there exists a
minimal qcb_{0} space X satisfying the equation X = A + [ B => X ]. This
least fixed point is obtained via a similar construction over a certain class of
domains with partial equivalence relations, using standard domain
representations of A and B. In fact, a least fixed point exists for every
strictly positive operator.
Further generalisations, e.g. admitting generalised positive induction or free
algebra constructions in the defining equation, tend to somewhat complicate the
situation.