 The motivation for the kinds of positive set theory we are familiar with is topological. We are to understand the sets as closed sets under some topology. Finite unions and intersections of closed sets are closed; this supports the inclusion of \(\{x \mid \phi \lor \psi \}\) and \(\{x \mid \phi \amp \psi \}\) as sets if \(\{x \mid \phi \}\) and \(\{x \mid \psi \}\) are sets. Arbitrary intersections of closed sets are closed: this supports our adoption of even bounded universal quantification (if each \(\{x \mid \phi(y)\}\) is a set, then \(\{x \mid \forall y\phi(y)\}\) is the intersection of all of these sets, and so should be closed, and \(\{x \in A \mid \forall y\phi(y)\}\) is also an intersection of closed sets and so should be closed. The motivation for permitting \(\{x \mid \exists y\phi(y)\}\) when each \(\{x \mid \phi(y)\}\) exists is more subtle, since infinite unions do not as a rule preserve closedness: the idea is that the set of pairs \((x, y)\) such that \(\phi(x, y)\) is closed, and the topology is such that the projection of a closed set is closed. Compactness of the topology suffices. Moreover, we now need to be aware that formulas with several parameters need to be considered in terms of a product topology.