Abstract
In a topological space that has a countable base, it is obvious that the union of any family of open sets of first category is of first category. One need only take the union of those members of the base that are contained in at least one member of the given family. The same reasoning shows that the union of any family of open sets of measure zero has measure zero (for any measure defined for all open sets). It is remarkable that the first statement remains valid whether the space has a countable base or not. The second statement, however, needs to be qualified.
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© 1980 Springer-Verlag New York Inc.
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Oxtoby, J.C. (1980). The Banach Category Theorem. In: Measure and Category. Graduate Texts in Mathematics, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9339-9_16
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DOI: https://doi.org/10.1007/978-1-4684-9339-9_16
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9341-2
Online ISBN: 978-1-4684-9339-9
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