Abstract
In this paper, we obtain results of the following type: if f : X → Y f:X \to Y is a closed map and X X is some "nice" space, and Y 2 {Y^2} is a k k -space or has countable tightness, then the boundary of the inverse image of each point of Y Y is "small" in some sense, e.g., Lindelöf or ω 1 {\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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