We show that the category of basic pairs (BP) and the category of concrete spaces (CSpa) are both small-complete and small-cocomplete in the framework of constructive Zermelo-Frankel set theory extended with the set generation axiom. We also show that CSpa is a coreflective subcategory of BP.
The uniform continuity theorem (UCT) states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, sans-serifUCT is strictly stronger than the decidable fan theorem (DFT), but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that sans-serifDFT is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that sans-serifDFT is equivalent to a similar principle for real‐valued functions on the Cantor space false{0,1false}N. These results extend Berger's [2] characterisation of sans-serifDFT for integer‐valued functions on false{0,1false}N and unify some characterisations of sans-serifDFT in terms of functions having continuous moduli.
Brouwer-operations, also known as inductively defined neighbourhood functions, provide a good notion of continuity on Baire space which naturally extends that of uniform continuity on Cantor space. In this paper, we introduce a continuity principle for Baire space which says that every pointwise continuous function from Baire space to the set of natural numbers is induced by a Brouweroperation. Working in Bishop constructive mathematics, we show that the above principle is equivalent to a version of bar induction whose strength is between that of the monotone bar induction and the decidable bar induction. We also show that the monotone bar induction and the decidable bar induction can be characterised by similar principles of continuity. Moreover, we show that the Π 0 1 bar induction in general implies LLPO (the lesser limited principle of omniscience). This, together with the fact that the Σ 0 1 bar induction implies LPO (the limited principle of omniscience), shows that an intuitionistically acceptable form of bar induction requires the bar to be monotone.
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