Stratifiability and the $\mu$-space property of function spaces with intermediate topologies
Keywords:
Stratifiable space, $M_3$-space, $M_1$-space, $\mu$-space, function space, topology of pointwise convergence, compact-open topologyAbstract
Dedicated to the memory of Prof. Phillip L. Zenor
We begin to investigate the stratifiability and the $\mu$ space property of intermediate topologies between the toplogies of $C_p(X)$ and $C_k(X)$, for a separable metrizable space $X$. In particular, for the space $\mathbb{P}$ of irrational numbers, we show the following:
(1) There is a family $\mathcal{K}$ of compact sets of $\mathbb{P}$ such that $C_{\mathcal{K}}(\mathbb{P})$ is an $M_1$-space and the topology of $C_{\mathcal{K}}(\mathbb{P})$ is strictly between that of $C_p(X)$ and that of $C_k(\mathbb{P})$.
(2) For any nonzero natural number $n$, let $\mathcal{K}$ be the family of all compact sets with scattered height $<n$. Then $C_{\mathcal{K}}(\mathbb{P})$ is neither a stratifiable space nor a $\mu$-space.
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