Products of 𝑘-spaces and spaces of countable tightness

Gruenhage, G.; Tanaka, Y.

doi:10.1090/S0002-9947-1982-0664043-9

Products of $k$-spaces and spaces of countable tightness
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Trans. Amer. Math. Soc. 273 (1982), 299-308 Request permission

Abstract:

In this paper, we obtain results of the following type: if $f:X \to Y$ is a closed map and $X$ is some "nice" space, and ${Y^2}$ is a $k$-space or has countable tightness, then the boundary of the inverse image of each point of $Y$ is "small" in some sense, e.g., Lindelöf or ${\omega _1}$-compact. We then apply these results to more special cases. Most of these applications combine the "smallness" of the boundaries of the point-inverses obtained from the earlier results with "nice" properties of the domain to yield "nice" properties on the range.

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