Continuous functions on compact totally ordered spaces
The authors prove that for any compact linearly ordered space \(K\), the space \(C_p(K)\) of continuous functions on \(K\) endowed with the topology of pointwise convergence is \(\sigma\)-fragmented using closed sets. The notion of \(\sigma\)-fragmentability was introduced by the same authors in [Mathematika 39, No. 1, 161-188 (1992; Zbl 0761.46008), No. 2, 197-215 (1992; Zbl 0761.46009)], where this was used to study various properties of (dual) Banach spaces. The above theorem combined with some results of Cornett, Pearson, van Mill and Wattel, and of Nikiel implies that \(C_p(K)\) is \(\sigma\)-fragmented when \(K\) is a dendron or a compact space with a closed crossfree subbase for its topology.
scientific article; zbMATH DE number 850621