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. | ||
A G<sub>δ</sub> in a [[complete metric space]] is again a complete metric space. | ==Properties== | ||
* The [[pre-image]] of a G<sub>δ</sub> set under a continuous map is again a G<sub>δ</sub> set. In particular, the [[zero set]] of a continuous real-valued function is a G<sub>δ</sub> set. | |||
* A closed G<sub>δ</sub> set is a [[normal space]] is the zero set of a continuous real-valued function. | |||
* A G<sub>δ</sub> in a [[complete metric space]] is again a complete metric space. | |||
==Gδ space== | ==Gδ space== | ||
A '''G<sub>δ</sub> space''' is a topological space in which every | A '''G<sub>δ</sub> space''' is a topological space in which every closed set is a G<sub>δ</sub> set. A [[normal space]] which is also a G<sub>δ</sub> space is '''[[perfectly normal space|perfectly normal]]'''. Every metrizable space is perfectly normal, and every perfectly normal space is a [[completely normal space]]; neither implication is reversible. | ||
==References== | ==References== | ||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=134,207-208 }} | * {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=134,207-208 }} | ||
* {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=162 }} | * {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=162 }} | ||
Revision as of 01:30, 4 January 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.