Methods of the theory of cardinal invariants and the theory of mappings as applied to spaces of functions (Q1814393)

Given a Tychonoff space \(X\), denote by \(C_ p(X)\) the set of all real valued continuous functions on \(X\) endowed with the pointwise topology. The author proves that the locally convex spaces (and in particular --- the spaces of type \(C_ p(X)\)) are not normal noncountable paracompact. It is shown also that the complete normality, the complete countable paracompactness and the perfect normality are equivalent properties for the spaces of type \(C_ p(X)\). Properties related with a given cardinal \(\tau\) and cardinal-valued functions, which generalize \(\tau\)- monotonicity and \(\tau\)-stability are considered. A duality theorem with respect to the functor \(C_ p\) is obtained. The invariantness of some of these properties with respect to continuous mappings between spaces of type \(C_ p(X)\) is proved. In particular if \(X\) is discrete and \(\phi:C_ p(X)\to C_ p(Y)\) is perfect, then \(C_ p(X)\) is homeomorphic to \(C_ p(Y)\). Theorem: If \(C_ p(X)\) is homeomorphic to a retract of a \(G_ \delta\)- set in \(\mathbb{R}^ \tau\) then \(X\) is discrete. Various corollaries and properties are given as well.