Abstract
Let S be a topological space(1) and let C(S) denote the set of all real valued, bounded, continuous functions on S. It is well known that C(S) is a distributive lattice under the operations sup (f,g) and inf (f,g). In general, however, C(S) is not a complete lattice; that is, an arbitrary bounded set of continuous functions in C(S) need not have a least upper bound in the lattice C(S). Furthermore, the structure of the minimal completion of C(S) by means of normal subsets has not been determined even in the simple case where S is the real interval [0, 1].
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Dilworth, R.P. (1990). The Normal Completion of the Lattice of Continuous Functions. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_40
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_40
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