JournalsrsmupVol. 129pp. 47–69

On the Functionally Countable Subalgebra of C(X)C(X)

  • Mostafa Ghadermazi

    University of Kurdistan, Sanandaj, Iran
  • O.A.S. Karamzadeh

    Chamran University, Ahvaz, Iran
  • M. Namdari

    Chamran University, Ahvaz, Iran
On the Functionally Countable Subalgebra of $C(X)$ cover
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Abstract

Let Cc(X)={fC(X):f(X) is countable}{C_c{{\char 40}}X{{\char 41}}} =\{\,f\in C(X): f(X) \hbox{ is countable}\}. Similar to C(X)C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring Cc(X){C_c{{\char 40}}X{{\char 41}}} is either Cc(X){C_c{{\char 40}}X{{\char 41}}} or a semiprime (resp. prime) ideal in Cc(X){C_c{{\char 40}}X{{\char 41}}} . For an ideal II in Cc(X){C_c{{\char 40}}X{{\char 41}}} , it is observed that II and I\sqrt{I} have the same largest zcz_c-ideal. If XX is any topological space, we show that there is a zero-dimensional space YY such that Cc(X)Cc(Y){C_c{{\char 40}}X{{\char 41}}} \cong {C_c{{\char 40}}Y{{\char 41}}} . Consequently, if XX has only countable number of components, then Cc(X)C(Y){C_c{{\char 40}}X{{\char 41}}} \cong C(Y) for some zero-dimensional space YY. Spaces X for which Cc(X){C_c{{\char 40}}X{{\char 41}}} is regular (called CPCP-spaces) are characterized both algebraically and topo log ically and it is shown that PP-spaces and CPCP-spaces coincide when XX is zero-dimensional. In contrast to C(X)C^*(X), we observe that Cc(X){C_c{{\char 40}}X{{\char 41}}} enjoys the algebraic properties of regularity, 0\aleph _{_0}-selfinjectivity and some others, whenever C(X)C(X) has these properties. Finally an example of a space XX such that Cc(X){C_c{{\char 40}}X{{\char 41}}} is not isomorphic to any C(Y)C(Y) is given.

Cite this article

Mostafa Ghadermazi, O.A.S. Karamzadeh, M. Namdari, On the Functionally Countable Subalgebra of C(X)C(X). Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 47–69

DOI 10.4171/RSMUP/129-4