Gδ space: Difference between revisions
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Undid revision 1214671836 by Samuel Adrian Antz (talk) The nlab article is about G-delta sets, not G-delta spaces, so maybe you meant to add the nlab external link to that other article instead? |
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{{Short description|Property of topological space}} |
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{{DISPLAYTITLE:G<sub>δ</sub> space}} |
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| ⚫ | In mathematics, particularly [[topology]], a '''G<sub>δ</sub> space''' is a [[topological space]] in which [[closed set]]s are in a way ‘separated’ from their complements using only countably many [[open set]]s. A G<sub>δ</sub> space may thus be regarded as a space satisfying a different kind of [[separation axiom]]. In fact [[normal space|normal]] G<sub>δ</sub> spaces are referred to as [[perfectly normal space]]s, and satisfy the strongest of [[separation axioms]]. |
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G<sub>δ</sub> spaces are also called '''perfect spaces'''.<ref>Engelking, 1.5.H(a), p. 48</ref> The term ''perfect'' is also used, incompatibly, to refer to a space with no [[isolated point]]s; see [[Perfect set]]. |
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==Terminology== |
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==Definition== |
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* If ''A'' is a subset of a [[topological space]], then ''A'' is said to be a [[G-delta set]] if ''A'' can be written as the countable intersection of open sets. Trivially, any open subset of a topological space is a G-delta set. |
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A countable [[intersection]] of open sets in a topological space is called a [[G-delta set|G<sub>δ</sub> set]]. Trivially, every open set is a G<sub>δ</sub> set. Dually, a countable union of closed sets is called an [[F-sigma set|F<sub>σ</sub> set]]. Trivially, every closed set is an F<sub>σ</sub> set. |
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A topological space ''X'' is called a '''G<sub>δ</sub> space'''<ref>Steen & Seebach, p. 162</ref> if every closed subset of ''X'' is a G<sub>δ</sub> set. Dually and equivalently, a ''G<sub>δ</sub> space'' is a space in which every open set is an F<sub>σ</sub> set. |
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==Formal definition== |
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If ''X'' is a topological space, then ''X'' is said to be a G-delta space if every closed subspace of ''X'' is a G-delta set. |
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* Every subspace of a G<sub>δ</sub> space is a G<sub>δ</sub> space. |
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* Every [[metrizable space]] is a G<sub>δ</sub> space. The same holds for [[pseudometrizable space]]s. |
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* Every [[second countable]] [[regular space|regular]] space is a G<sub>δ</sub> space. This follows from the [[Urysohn's metrization theorem]] in the Hausdorff case, but can easily be shown directly.<ref>{{Cite web|url=https://math.stackexchange.com/q/1966730|title=General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem}}</ref> |
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* Every countable regular space is a G<sub>δ</sub> space. |
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* Every [[hereditarily Lindelöf]] regular space is a G<sub>δ</sub> space.<ref>https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1</ref> Such spaces are in fact [[normal space|perfectly normal]]. This generalizes the previous two items about second countable and countable regular spaces. |
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* A G<sub>δ</sub> space need not be normal, as '''R''' endowed with the [[K-topology]] shows. That example is not a regular space. Examples of [[Tychonoff space|Tychonoff]] G<sub>δ</sub> spaces that are not normal are the [[Sorgenfrey plane]]<ref>{{Cite web|url=https://dantopology.wordpress.com/2014/05/07/the-sorgenfrey-plane-is-subnormal/|title = The Sorgenfrey plane is subnormal|date = 8 May 2014}}</ref> and the [[Niemytzki plane]].<ref>{{Cite web|url=https://math.stackexchange.com/q/2711463|title=General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces}}</ref> |
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* In a [[first countable]] [[T1 space|T<sub>1</sub> space]], every [[singleton (mathematics)|singleton]] is a G<sub>δ</sub> set. That is not enough for the space to be a G<sub>δ</sub> space, as shown for example by the [[lexicographic order topology on the unit square]].<ref>{{Cite web|url=https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space/|title=The Lexicographic Order and the Double Arrow Space|date=8 October 2009}}</ref> |
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* The [[topological sum]] <math>X={\coprod}_i X_i</math> of a family of disjoint topological spaces is a G<sub>δ</sub> space if and only if each <math>X_i</math> is a G<sub>δ</sub> space. |
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==Notes== |
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{{reflist}} |
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* In G-delta spaces, every open set is the countable union of closed sets. In fact, a topological space is a G-delta space iff every open set is an [[F-sigma set]] |
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* Any [[metric space]] is a G-delta space. |
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* Without assuming Urysohn’s metrization theorem, one can prove that every [[regular space]] with a [[Second countable space|countable base]] is a G-delta space. |
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* A G-delta space need not be normal as '''R''' endowed with the [[K-topology]] shows. |
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* In a [[First countable space|first countable]] T<sub>1</sub> space, any one point set is a G-delta set. |
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==See also== |
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* [[Separation axioms]] |
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* [[G-delta set]] |
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* [[F-delta set]] |
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* [[Normal space]] |
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* [[Metric space]] |
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==References== |
==References== |
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* {{cite book|last=Engelking|first=Ryszard| authorlink=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}} |
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* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover Publications reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}} |
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* Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". ''The American Mathematical Monthly'', Vol. 77, No. 2, pp. 172–176. [https://www.jstor.org/stable/2317335 on JStor] |
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{{DEFAULTSORT:Gdelta Space}} |
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[http://www.jstor.org/pss/2317335 A compact non-metrizable space such that every closed set is a G-delta set] |
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[[Category:General topology]] |
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[[Category:Properties of topological spaces]] |
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[[Category:Real analysis]] |
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Latest revision as of 03:35, 21 March 2024
In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
Gδ spaces are also called perfect spaces.[1] The term perfect is also used, incompatibly, to refer to a space with no isolated points; see Perfect set.
Definition
[edit]A countable intersection of open sets in a topological space is called a Gδ set. Trivially, every open set is a Gδ set. Dually, a countable union of closed sets is called an Fσ set. Trivially, every closed set is an Fσ set.
A topological space X is called a Gδ space[2] if every closed subset of X is a Gδ set. Dually and equivalently, a Gδ space is a space in which every open set is an Fσ set.
Properties and examples
[edit]- Every subspace of a Gδ space is a Gδ space.
- Every metrizable space is a Gδ space. The same holds for pseudometrizable spaces.
- Every second countable regular space is a Gδ space. This follows from the Urysohn's metrization theorem in the Hausdorff case, but can easily be shown directly.[3]
- Every countable regular space is a Gδ space.
- Every hereditarily Lindelöf regular space is a Gδ space.[4] Such spaces are in fact perfectly normal. This generalizes the previous two items about second countable and countable regular spaces.
- A Gδ space need not be normal, as R endowed with the K-topology shows. That example is not a regular space. Examples of Tychonoff Gδ spaces that are not normal are the Sorgenfrey plane[5] and the Niemytzki plane.[6]
- In a first countable T1 space, every singleton is a Gδ set. That is not enough for the space to be a Gδ space, as shown for example by the lexicographic order topology on the unit square.[7]
- The Sorgenfrey line is an example of a perfectly normal (i.e. normal Gδ) space that is not metrizable.
- The topological sum of a family of disjoint topological spaces is a Gδ space if and only if each is a Gδ space.
Notes
[edit]- ^ Engelking, 1.5.H(a), p. 48
- ^ Steen & Seebach, p. 162
- ^ "General topology - Every regular and second countable space is a $G_\delta$ space, without assuming Urysohn's metrization theorem".
- ^ https://arxiv.org/pdf/math/0412558.pdf, lemma 6.1
- ^ "The Sorgenfrey plane is subnormal". 8 May 2014.
- ^ "General topology - Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces".
- ^ "The Lexicographic Order and the Double Arrow Space". 8 October 2009.
References
[edit]- Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4.
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover Publications reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
- Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". The American Mathematical Monthly, Vol. 77, No. 2, pp. 172–176. on JStor