Continuous functions with compact support

Sudip Kumar Acharyya, K. C. Chattopadhyaya, Partha Pratim Ghosh

Abstract

The main aim of this paper is to investigate a subring of the ring of continuous functions on a topological space X with values in a linearly ordered field F equipped with its order topology, namely the ring of continuous functions with compact support. Unless X is compact, these rings are commutative rings without unity. However, unlike many other commutative rings without unity, these rings turn out to have some nice properties, essentially in determining the property of X being locally compact non-compact or the property of X being nowhere locally compact. Also, one can associate with these rings a topological space resembling the structure space of a commutative ring with unity, such that the classical Banach Stone Theorem can be generalized to the case when the range field is that of the reals.


Keywords

Ordered Fields; Zero Dimensional Spaces; Strongly Zero Dimensional Spaces; Compactifications

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References

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S. K. Acharyya, K. C. Chattopadhyaya and P. P. Ghosh, Constructing Banaschewski Compactification Without Dedekind Completeness Axiom, to appear in International Journal for Mathematics and Mathematical Sciences.

S. K. Acharyya, K. C. Chattopadhyaya and P. P. Ghosh, The rings Ck(X) and C∞(X), some remarks, Kyungpook Journal of Mathematics, 43 (2003), 363 - 369.

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