Date: Fri, 26 Jul 2002 15:56:09 +0200 (MEST) From: Philippe Gaucher <gaucher@math.u-strasbg.fr> Hello,
Philippe Gaucher wrote:
In the category CGTop of compactly generated Hausdorff topological spaces, the product provides a symmetric closed monoidal structure.
Question: Is there another symmetric closed monoidal structure on CGTop ?
Yes, There is at least one more: the "componentwise" structure like in Top. Define the tensor product X#Y as the cartesian product endowed with the final topology with respect to all maps X->X#Y, x|->(x,y) for all (fixed) y in Y and all maps Y->X#Y, y|->(x,y) for all (fixed) x in X. Then X#Y is compactly generated because cg spaces are closed under final structures. Moreover, X#Y is Hausdorff because its topology is finer than the product topology. Obviously, this gives a symmetric monoidal structure on CGTop. A map X#Y->Z is continuous, iff it is separately continuous (i.E. contiuous in each variable). In order to define function spaces, first proceed as in Top; i.e for spaces X,Y consider the set of all continiuous maps from X to Y with the topology of pointwise convergence, i.e the topology as a subspace of the X-th cartesian power of Y. Unfortunately, this topology need not be compactly generated, but if we switch to its cg modification, everything works, and we obtain a symmetric monoidal closed structure. This structur does not coincide with the cartesian structure because there are separately continuos but non-continuous real functions on the product of the real line with itself. I do not know whether there are more than two symmetric monoidal structures on CGTop. Greetings Reinhard 29-Jul-2002 17:09:04 -0300,1556;000000000001-00000028