Projective objects in categories of locally convex spaces (Q1093147)

The aim of this paper is to determine the projective objects in the prevariety of locally convex spaces (i.e. a subcategory of the category of locally convex spaces, closed under passage to subspaces and products). Let the categories of (1) Tychonoff spaces (with continuous mappings as morphisms) and (2) Hausdorff locally convex spaces (with linear continuous maps as morphisms) be denoted by (1) HW and (2) HLC. PR(K) denotes the class of projective objects. The subcategory K will be assumed to be complete and saturated. Then the main result of this note is the following: Theorem (a) Let K be a prevariety in HLC. Then \(X\in PR(K)\) iff \(X=F(K,Z)\), where Z is a discrete space and F(K,Z) is the free locally convex space with Hamel basis Z such that injection of Z into F(K,Z) is a homeomorphism. (b) Let \(W_ 1\) be a subcategory of HW such that \(Obj(HLC)\subseteq Obj(W_ j)\). Then \(X\in PR(W_ 1)\) iff X is discrete.