“School of Mathematics”
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| Paper IPM / M / 2944 |
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| Abstract: | |||||
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The spaces X in which every prime z°-ideal of C(X) is
either minimal or maximal are characterized. By this
characterization, it turns out that for a large class of
topological spaces X, such as metric spaces, basically
disconnected spaces and one-point compactification of discrete
spaces, every prime z°-ideal in C(X) is either minimal or
maximal. We will also answer the following questions: When is
every nonregular prime ideal in C(X) a z°-ideal? When is
every nonregular (prime) z-ideal in C(X) a z°-ideal? For
instance, we have shown that every nonregular prime ideal of
C(X) is a z°-ideal if and only if X is ∂-space
(a space in which the boundary of any zeroset is again a zeroset).
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