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Paper IPM / M / 7408  


Abstract:  
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 X ≅ Y. 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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