Supports of continuous functions
TLDR
In this paper, it was shown that every real-compact space with pseudocompact support has compact support and every i/i compact space is /?compact.Abstract:
Gillman and Jerison have shown that when A" is a realcompact space, every function in C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called fi-compact. In this paper we give several characterizations of /?-compact spaces and also introduce and study a related class of spaces, the ^-compact spaces ; these are spaces X with the property that every function in C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ^-compact and every i/i-compact space is /?-compact. A family & of subsets of a space X is said to be stable if every function in C(X) is bounded on some member of #". We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.read more
Citations
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Journal ArticleDOI
On the intrinsic topology and some related ideals of $C(X)$
O. A. S. Karamzadeh,M. Rostami +1 more
TL;DR: In this paper, the authors studied the intrinsic topology on a completely regular Hausdorf space X and derived the minimal ideals and the socle of C(X) via their corresponding z-filters.
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Essential ideals inC(X)
TL;DR: The Frechet z-filter is the intersection of essential z-filters as mentioned in this paper, and it is shown that if every essential ideal inC(X) is a z-ideal then X is a P-space, if and only if every open set containing all nonisolated points is cofinite.
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Maximal ideals in subalgebras of c(x)
Lothar Redlin,Saleem Watson +1 more
TL;DR: In this article, the maximal ideals of continuous real-valued functions on a completely regular space X and its subalgebra C*(X) of bounded functions have been studied extensively.
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Functions with pseudocompact support
D. G. Johnson,Mark Mandelker +1 more
TL;DR: In this paper, the authors characterized the ideal structure of C(X) and the topological structure of βX in terms of μ-, η-, and ψ-compactifications.
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Prime and maximal ideals in subrings of C(X)
H. Linda Byun,Saleem Watson +1 more
TL;DR: In this article, the authors studied the prime and maximal ideals in subrings A(X) of C(X), and showed that many of the results known separately for C(x) and C∗(x), often by different methods, are true for any such subring.
References
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Rings of Continuous Functions.
Book
Rings of continuous functions
TL;DR: In this paper, the Stone-Czech Compactification is used to define a topological space, and a list of symbols for topological spaces is presented, including cardinal of closed sets in Beta-x, homomorphisms and continuous mapping.
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Rings of real-valued continuous functions. I
TL;DR: In this paper, the authors studied the ring of all real-valued continuous functions (X, IR) on a tpopological space X, and showed that these functions can be represented by hyperreal fields (H-fields).
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On Two Problems of Urysohn
TL;DR: In this paper, it was shown that a regular Hausdorff space in which every continuous real-valued function is constant and which is accordingly connected can be constructed in several steps.
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On the Hewitt realcompactification of a product space
TL;DR: Hager as mentioned in this paper showed that the relation 3(X x Y) = gX x / Y holds if and only if X x Yc vX is pseudocompact.