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arXiv · 1612.05410

Topological aspects of order in $C(X)$

Marko Kandić·Aleš Vavpetič
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Abstract

In this paper we consider the relationship between order and topology in the vector lattice $C_b(X)$ of all bounded continuous functions on a Hausdorff space $X$. We prove that the restriction of $f\in C_b(X)$ to a closed set $A$ induces an order continuous operator iff $A=\overline{\mathrm{Int} A}.$ This result enables us to easily characterize bands and projection bands in $C_0(X)$ and $C_b(X)$ through the one-point compactification and the Stone-Čech compactification of $X$, respectively. With these characterizations we describe order complete $C_0(X)$ and $C_b(X)$-spaces in terms of extremally disconnected spaces. Our results serve us to solve an open question on lifting un-convergence in the case of $C_0(X)$ and $C_b(X)$.

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Marko Kandić, Aleš Vavpetič. 2016-12-16. Topological aspects of order in $C(X)$. https://doi.org/10.1007/s11117-018-0628-8

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