The principle of pointfree continuity

TL;DR

This paper develops a constructive pointfree topology framework to formalize a principle of continuity, linking it to spatiality, bar induction, and compactness within the Minimalist Foundation, offering new insights into constructive topology.

Contribution

It introduces a notion of continuous operation between pointfree topologies and establishes a principle of pointfree continuity, connecting it to spatiality and classical principles in constructive topology.

Findings

The principle holds under bi-spatiality of the source topology.

For formal Baire space and unit interval, the principle equates to their spatiality.

Recasts connections between spatiality, bar induction, and compactness in terms of the new principle.

Abstract

In the setting of constructive pointfree topology, we introduce a notion of continuous operation between pointfree topologies and the corresponding principle of pointfree continuity. An operation between points of pointfree 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 pointfree continuity for pointfree topologies $\mathcal{S}$ and $\mathcal{T}$ says that any relation which induces a continuous operation between points is a morphism from $\mathcal{S}$ to $\mathcal{T}$. The principle holds under the assumption of bi-spatiality of $\mathcal{S}$. When $\mathcal{S}$ is the formal Baire space or the formal unit interval and $\mathcal{T}$ is the formal topology of natural numbers, the principle is equivalent to spatiality of the formal Baire space and formal unit interval, respectively. Some of the well-known connections between spatiality, bar induction, and compactness of the unit interval are recast in terms of our principle of continuity. We adopt the Minimalist Foundation as our constructive foundation, and positive topology as the notion of pointfree 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.