Productivity of Zariski-compactness for constructs of affine spaces
In this paper an affine space is simply a set \(X\) together with a family of subsets which contains both the empty set and the set \(X\). A morphism of such spaces is a function for which the inverse image maps the defining subsets to the corresponding defining subsets. Together these form the topological category SSETS. Associated with this category is a closure operator \(z\) (Zariski closure) where the \(z\)-closure, \(z(M)\), of an embedding \(M\to X\) is the subset of \(X\) consisting of all \(x\) in \(X\) for which all pairs of maps \(X\to S\) which agree on \(M\) also agree at \(x\). Here \(S= (\{0,1\}; \{0,\{1\},\{0,1\}\})\) is the Sierpiński affine space. An object \(X\) in a non-trivial hereditary concretely coreflective subconstruct, \(C\), of SSETS is said to be \(z\)-compact if the second factor projection from the product, \(X\times Y\) in \(C\), to \(Y\) is \(z\)-preserving (i.e. commutes with \(z\)-closure) for all objects \(Y\) in \(C\). The main result of the paper is: If in \(C\) arbitrary products of quotients are quotients, then \(z\)-compactness is productive in \(C\) if and only if either all \(z\)-compact objects are indiscrete or \(z\)-compactness coincides with \(z\)-completeness on the subcategory of \(T_0\) objects in \(C\). Several examples of well known subconstructs of SSETS are considered. In particular it is shown that \(z\)-compactness is productive in SSETS and in CL, the category of closure spaces.
scientific article; zbMATH DE number 5013550
scientific article