We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

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Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.