G-delta set: Difference between revisions
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In [[general topology]], a '''G<sub>δ</sub> set''' is a [[subset]] of a [[topological space]] which is a [[countability|countable]] [[intersection]] of [[open set]]s. An '''F<sub>σ</sub>''' space is similarly a countable [[union]] of [[closed set]]s. | In [[general topology]], a '''G<sub>δ</sub> set''' is a [[subset]] of a [[topological space]] which is a [[countability|countable]] [[intersection]] of [[open set]]s. An '''F<sub>σ</sub>''' space is similarly a countable [[union]] of [[closed set]]s. | ||
Revision as of 13:16, 8 March 2009
In general topology, a Gδ set is a subset of a topological space which is a countable intersection of open sets. An Fσ space is similarly a countable union of closed sets.
Properties
- The pre-image of a Gδ set under a continuous map is again a Gδ set. In particular, the zero set of a continuous real-valued function is a Gδ set.
- A closed Gδ set is a normal space is the zero set of a continuous real-valued function.
- A Gδ in a complete metric space is again a complete metric space.
Gδ space
A Gδ space is a topological space in which every closed set is a Gδ set. A normal space which is also a Gδ space is perfectly normal. Every metrizable space is perfectly normal, and every perfectly normal space is a completely normal space; neither implication is reversible.
References
- J.L. Kelley (1955). General topology. van Nostrand, 134,207-208.
- Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 162. ISBN 0-387-90312-7.