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At(x At(y attention at stage basic open sets bicontinuous classical closed interval closed unit interval complement components of G computable homeomorphism contains contradiction converges corresponding dense set disconnected effective uniform continuity effectively uniformly Euclidean 2-space finite SIB formulae intersect irreflexive lattice of r.e. least element linear ordering lower cut modulus of convergence natural numbers non-cut points open ball order arithmetic order topology ordering with endpoints Proof of Lemma pseudo-recursive r.e. closed sets r.e. closed subsets r.e. open sets radius ran(g rational point receives attention recursive points recursive real relativized satisfying WC segments semicircle of recursive sets of recursive subsets of Euclidean subsets of recursive Suppose theorem theory of RR tolerance square topological space true in RR undecidable uniformly continuous unit interval Vx(At(x x-axis x-coordinate X(RR