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Keywords:
quasi $P$-space; $P$-space; scattered space; Cantor-Bendixson derivatives; \newline nodec space; quasinormality
quasi $P$-space; $P$-space; scattered space; Cantor-Bendixson derivatives; \newline nodec space; quasinormality
Summary:
Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C(X)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset is a $z$-embedded $P$-space, then $X$ is a quasi $P$-space. Conversely, if $X$ is a quasi $P$-space and $F$ is a nowhere dense $z$-embedded zeroset, then $F$ is a $P$-space. On the other hand, there are examples of countable quasi $P$-spaces with no $P$-points at all. If a product $X\times Y$ is normal and quasi $P$, then one of the factors must be a $P$-space. Conversely, if one of the factors is a compact quasi $P$-space and the other a $P$-space then the product is quasi $P$. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^{-1}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. The converse fails even with more stringent assumptions on the map $f$. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi $P$-spaces is always quasi $P$.
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