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B: b®t). It is easy to verify that this is a product. 6 Proposition If a category C has pullbacks for every pair of arrows and it has terminal object, then it has an equalizer for every pair of arrows. Proof Let f,g: a®b. Let (c, fst:c®a, snd:c®a) be the pullback of (: a®b´a, : a®b´a). Then the equalizer of f,g is (c, fst = snd). Indeed, f û fst = p1û = p1û û fst = p2 û û snd = p2 û = g û snd. Moreover, for any (c', h:c'®a) such that f û h = g û h , also û h = û h; by definition of pullback, there exists a unque k: c'®c such that fst û k = h.

Sets. ) These limits are computable in a very sound sense. e. sets. Exercise Prove a similar fact for the set P of partial maps from w to w . Call constructive domain a domain whose elements are the computable elements in an effectively given domain. Since ED is Cartesian closed, this may be done in any (higher) type. In particular, given the constructive domains Xc, Yc obtained from X and Y , one may consider the constructive domain YXc of the computable elements of YX . 6 CD is the category of constructive domains and continuous and computable morphisms.

From now on we assume that each countably based L-spaces (X,¯) comes with a given surjective enumeration e: w®Xo of the base. An immediate example of separable L-spaces is the set of real numbers endowed with the usual notion of sequence converge (Cauchy). , fÎ L[X,Y] iff " x Î X " {x i} ¯ x {f (xi )}¯ f (x), where convergence is given in the intended spaces. This category has exponents and products, as (L[X,Y],¯) also is an L-space by {fi}¯f iff "xÎX, "{xi}¯x {fi(xi)}¯f(x) , while products are given by componentwise convergence.