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On a question of C c ( X )

A. R. Olfati — 2016

Commentationes Mathematicae Universitatis Carolinae

In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of C ( X ) , Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that C c ( X ) is isomorphic to some ring of continuous functions if and only if υ 0 X is functionally countable. For a strongly zero-dimensional space X , this is equivalent to say that X is functionally countable. Hence for every P -space it is equivalent to pseudo- 0 -compactness.

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati — 2016

Commentationes Mathematicae Universitatis Carolinae

Let X be a zero-dimensional space and C c ( X ) be the set of all continuous real valued functions on X with countable image. In this article we denote by C c K ( X ) (resp., C c ψ ( X ) ) the set of all functions in C c ( X ) with compact (resp., pseudocompact) support. First, we observe that C c K ( X ) = O c β 0 X X (resp., C c ψ ( X ) = M c β 0 X υ 0 X ), where β 0 X is the Banaschewski compactification of X and υ 0 X is the -compactification of X . This implies that for an -compact space X , the intersection of all free maximal ideals in C c ( X ) is equal to C c K ( X ) , i.e., M c β 0 X X = C c K ( X ) . By applying methods of functionally...

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