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First Category


A subset E of a topological space S is said to be of first category in S if E can be written as the countable union of subsets which are nowhere dense in S, i.e., if E is expressible as a union

 E= union _(n in N)E_n

where each subset E_n subset S is nowhere dense in S. Informally, one thinks of a first category subset as a "small" subset of the host space and indeed, sets of first category are sometimes referred to as meager. Sets which are not of first category are of second category.

An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.

The rational numbers are of first category and the irrational numbers are of second category in R with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the set Z of integers with the subset topology inherited from R is (vacuously) of second category relative to itself because every subset of Z is open in Z with respect to that topology; on the other hand, Z is of first category in R with its standard topology and in Q with the subset topology inherited by Q from R. Likewise, the Cantor set is a Baire space (i.e., each of its open sets are of second category relative to it) even though it is of first category in the interval [0,1] with the usual topology.

Baire Category Theorem, Meager Set, Nowhere Dense, Nonmeager Set, Residual Set, Second Category


Portions of this entry contributed by Barnaby Finch

Portions of this entry contributed by Christopher Stover

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References

Hocking, J. and Young, G. Topology. New York: Dover, p. 89, 1961.Morgan, J. C. Point Set Theory. Boca Raton, FL: CRC Press, p. 21, 1989.Munkres, J. R. Topology: A First Course. Upper Saddle River, NJ: Prentice-Hall, pp. 293-294, 1975.Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

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First Category

Cite this as:

Finch, Barnaby; Stover, Christopher; and Weisstein, Eric W. "First Category." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstCategory.html

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