G-delta set: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(added section Properties)
mNo edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{subpages}}
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.


Line 11: Line 12:
==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 }}[[Category:Suggestion Bot Tag]]

Latest revision as of 17:00, 19 August 2024

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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