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Paper   IPM / M / 7408
School of Mathematics
  Title:   Rings of continuous functions vanishing at infinity
1.  A. R. Aliabad
2.  F. Azarpanah
3.  M. Namdari
  Status:   Published
  Journal: Comment. Math. Univ. Carolin.
  Vol.:  45
  Year:  2004
  Pages:   519-533
  Supported by:  IPM
We prove that a Hausdorff space X is locally compact if and only if its topology coincides with the weak topology induced by C(X). It is shown that for a Hausdorff space X, there exists a locally compact Hausdorff space Y such that C(X) ≅ C(Y). It is also shown that for locally compact spaces X and Y, C(X) ≅ C(Y) if and only if XY. Prime ideals in C(X) are uniquely represented by a class of prime ideals in C*(X). ∞-compact spaces are introduced and it turns out that a locally compact space X is ∞-compact if and only if every prime ideal in C(X) is fixed. The existence of the smallest ∞-compact space in βX containing a given space X is proved. Finally some relations between topological properties of the space X and algebraic properties of the ring C(X) are investigated. For example we have shown that C(X) is a regular ring if and only if X is an ∞-compact P-space.

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