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Brouwer-operations, also known as inductively defined neighbourhoodfunctions, provide a good notion of continuity on Baire space which naturallyextends that of uniform continuity on Cantor space. In this paper, we introducea continuity principle for Baire space which says that every pointwisecontinuous function from Baire space to the set of natural numbers is inducedby a Brouwer-operation. Working in Bishop constructive mathematics, we showthat the above principle is equivalent to a version of bar induction whosestrength is between that of the monotone bar induction and the decidable barinduction. We also show that the monotone bar induction and the decidable barinduction can be characterised by similar principles of continuity. Moreover,we show that the $\Pi^{0}_{1}$ bar induction in general implies LLPO (thelesser limited principle of omniscience). This, together with a fact that the$\Sigma^{0}_{1}$ bar induction implies LPO (the limited principle ofomniscience), shows that an intuitionistically acceptable form of bar inductionrequires the bar to be monotone.

Principles of bar induction and continuity on Baire space