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F_sigma Set


A subset F subset R of the real numbers is said to be an F_sigma set provided F is the countable union of closed sets. The name F_sigma comes from French: The F stands for fermé, meaning "closed," while the sigma stands for somme, meaning "sum."

This definition is dual to that of a Gdelta set in the sense that F is an F_sigma set if and only if its complement F^c=R-F is a G_delta.

Note that both F_sigma and G_delta sets are fundamental to measure theory. In particular, both types of sets are Borel sets and are considered to be the second level of the so-called Borel hierarchy.


See also

Borel Hierarchy, Borel Set, Closed Set, Gdelta Set, Open Set

This entry contributed by Christopher Stover

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References

Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.

Cite this as:

Stover, Christopher. "F_sigma Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/F-SigmaSet.html

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