“School of Mathematics”
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| Paper IPM / M / 7408 |
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| Abstract: | |||||||
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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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