Hereditarily $\sigma$-metacompact function spaces
Keywords:
function space, pointwise topology, k-network, $\sigma$-metacompact, supercompact spaceAbstract
Let $X$ be a compact space. We consider conditions on $X$ which ensure that the function space $C_p(X)$ is hereditarily $\sigma$-metacompact. The conditions involve $k$-networks with special properties. We prove that $C_p(X)$ is hereditarily $\sigma$-metacompact provided that either $X$ has a $\sigma$-pointwise-closure-preserving closed $k$-network or $X$ is supercompact.
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