The principle of point-free continuity
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In the setting of constructive point-free topology, we introduce a notion of continuous operation between point-free topologies and the corresponding principle of point-free continuity. An operation between points of point-free topologies is continuous if it is induced by a relation between the bases of the topologies; this gives a rigorous condition for Brouwer's continuity principle to hold. The principle of point-free continuity says that any relation which induces a continuous operation between points is a morphism between the corresponding point-free topologies. The principle holds under the assumption of bi-spatiality. When restricted to a certain class of morphisms from the formal Baire space and the formal unit interval, the principle is equivalent to spatiality of these point-free spaces. Some of the well-known connections between spatiality, bar induction, and compactness of the unit interval are also reviewed. We adopt the Minimalist Foundation as our constructive foundation, and positive topology as the notion of point-free topology. This allows us to distinguish ideal objects from constructive ones, and in particular, to interpret choice sequences as points of the formal Baire space.
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