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7. Question. Cont is preserved under Mer quotients and finite Mer sums. 5, a quotient of a sum of contigual spaces has to be locally totally bounded. Must a quotient of a locally contigual space be locally contigual? 8. Counterexample. Consider the Cech-Stone compactification of~. the natural numbers. Let p e SIN - IN and let X be the Top subspace of ~N with underlying set IN v {p}. Then X, being topological, is necessarily grilldetermined, but X does not lie in CH(Cont). In fact, the CH(Cont) coreflection of X is discrete.

Theorem 1. For any regular cardinal a ;;. K 1, the a-filtered co limits corronute in IB with the a-limits ; moreover, they are preserved and reflected by the closed unit ball functor U : lE + Sets. The a-filtered colimit is computed at the level of underlying vector spaces and provided with the semi-norm II [x]ll inf II y\I. yE[x] In the case of1B and JN, this is a norm because a;;. ~i. For the same reason, the forgetful functor u : lE + Sets preserves this colimi t. The theorem follows from the connnutation in Sets and proposition 4.

The natural numbers. Let p e SIN - IN and let X be the Top subspace of ~N with underlying set IN v {p}. Then X, being topological, is necessarily grilldetermined, but X does not lie in CH(Cont). In fact, the CH(Cont) coreflection of X is discrete. The proof is simple if one observes that any uniformly continuous map f:Y + X of a contigual merotopic space Y into X has finite range fY. 5 cannot be omitted. 9. Problem. Find an internal characterization of Top("\ CH(Cont) and of Near("I CH(Cont). Qt' the rationals with the usual topology, does not lie in Top nCH(Cont).

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