Abstract
The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C p (X) and C(X) are determined. The metrizability of CM p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM p (X) are given.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.
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Okhezin, D.S. Study of families of monotone continuous functions on Tychonoff spaces. J Math Sci 144, 4152–4183 (2007). https://doi.org/10.1007/s10958-007-0259-2
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DOI: https://doi.org/10.1007/s10958-007-0259-2