زیرجبر تفکیک پذیر تابعی C(X)

The useful role of C_c(X) in studying C(X) motivated us to introduce and study the functionally separable subalgebra C_{cd}(Y) of C(X). Let Y be a dense subset of X, C_{cd}(Y)={fin C(X): |f(Y)|leq {aleph}_۰}. Clearly, C_c(X)subseteq C_{cd}(Y)subseteq C(X) and C_{cd}(Y) behaves like C(X) and C_c(X) in more properties. If X is a functionally countable or separable space then C_{cd(Y)=C(X), in this case X is called functionally separable space. Whenever X is pseudocompact and beta X is separable, then each fin C(X) is countable on a dense subset of X. Conversely, if each fin C(X) is countable on a dense subset of X and each G_{delta} -set has nonempty interior, then C(X)=C_c(X). Locally functionally separable subalgebra of C(X) is denoted by C_{cod}(X) where C_{cod}(X)={fin C(X) : |f(Y)|leq aleph_۰ , text{~~for some open dense subset Y of X}}, clearly C_{cod}(X)subseteq L_c(X). For a locally compact and pseudocompact space X, C_{cod}X)=C(X) if and only if C_{cod}(beta X)=C(beta X). We introduce z_{cod}-ideals in C_{cod}(X) and trivially observe that most of the facts related to z-ideals are extendable to z_{cod}-ideals.

Keywords:

شمارا تابعی , تفکیک پذیر , تفکیک پذیری تابعی , تفکیک پذیر تابعی موضعی