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10 R. Meyerdefines the topology of uniform convergence on compact subsets on Hom.A; B/ becausethe latter is equicontinuous.There is a distinguished element in Hom.A; B/ as well, namely, the zero homomorphismA ! 0 ! B. Thus Hom.A; B/ becomes a pointed topological space.Proposition 23. The above construction provides an enrichment of C alg over thecategory of pointed Hausdorff topological spaces.Proof. It is clear that 0 ı f D 0 and f ı 0 D 0 for all morphisms f . Furthermore, wemust check that composition of morphisms is jointly continuous. This follows fromthe equicontinuity of Hom.A; B/.This enrichment allows us to carry over some important definitions from categoriesof spaces to C alg. For instance, a homotopy between two -homomorphismsf 0 ;f 1 W A ! B is a continuous path between f 0 and f 1 in the topological spaceHom.A; B/. In the following proposition, Map C .X; Y / denotes the space of morphismsin the category of pointed topological spaces, equipped with the compact-opentopology.Proposition 24 (compare Proposition 3.4 in [28]). Let A and B be C -algebras andlet X be a pointed compact space. ThenMap C X; Hom.A; B/ Š Hom A; C 0 .X; B/ as pointed topological spaces.Proof. If we view A and B as pointed topological spaces with the norm topology andbase point 0, then Hom.A; B/ Map C .A; B/ is a topological subspace with the samebase point because Hom.A; B/ also carries the compact-open topology. Since X iscompact, standard point set topology yields homeomorphismsMap C X; Map C .A; B/ Š Map C .X ^ A; B/Š Map C A; Map C .X; B/ D Map C A; C 0 .X; B/ :These restrict to the desired homeomorphism.In particular, a homotopy between two -homomorphisms f 0 ;f 1 W A B is equivalentto a -homomorphism f W A ! C.Œ0; 1; B/ with ev t ı f D f t for t D 0; 1,where ev t denotes the -homomorphismev t W C.Œ0; 1; B/ ! B;f 7! f.t/:We also haveC 0 X; C 0 .Y; A/ Š C 0 .X ^ Y; A/for all pointed compact spaces X, Y and all G-C -algebras A. Thus a homotopybetween two homotopies can be encoded by a -homomorphismA ! C Œ0; 1; C.Œ0; 1; B/ Š C.Œ0; 1 2 ;B/: