There are some very elementary proofs concerning the scope of categories associated with the category of topological spaces. Many categories have injective objects, but their properties depend on what families of subobjects are allowed. In the case of topological T_0-spaces, for example, two alternatives suggest themselves: (1) arbitrary subspaces, and (2) dense subspaces. Both notions are interesting. Thus, a space D is injective in sense (1) iff for any space Y and subspace X and any continuous function f:X-->D there is a continuous extension f':Y-->D of f. The two-point T_0-space with one open point and one closed point is obviously injective by the very definition of "subspace". Injective spaces are also easily proved to be closed under arbitrary products and continuous retracts, which facts provide many other examples. Perhaps it is not so obvious, however, that injective spaces are also closed under the formation of function spaces, once the space of continuous functions is given the right topology; indeed the category of injective spaces and continuous functions is a cartesian closed category. Many of the properties of these spaces are provable once the spaces can be characterized as a kind of complete lattice with an appropriate, uniquely determined topology; continuity of functions then comes down to preservation of sups of directed subsets of the lattice. Alternatively, all T_0-spaces are embeddable in powerset spaces. As the function space between two powersets proves to be a retract of a suitable powerset, the desired properties follow. The talk will review old and new results about injective spaces, applications of the results, and a recent use of injectives to define a cartesian closed extension of the category of all T_0-spaces. This topological point of view makes it obvious that lambda-calculus models exist in many forms. |

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