Editing Subspace topology

{{Short description|Inherited topology}}
{{Redirect|Induced topology|the topology generated by a family of functions|Initial topology}}

In [[topology]] and related areas of [[mathematics]], a '''subspace''' of a [[topological space]] (''X'', ''𝜏'') is a [[subset]] ''S'' of ''X'' which is equipped with a [[Topological space#Definitions|topology]] induced from that of ''𝜏'' called the '''subspace topology'''<ref name=ttd>{{citation
 | last = tom Dieck | first = Tammo
 | doi = 10.4171/048
 | isbn = 978-3-03719-048-7
 | mr = 2456045
 | page = 5
 | publisher = European Mathematical Society (EMS), Zürich
 | series = EMS Textbooks in Mathematics
 | title = Algebraic topology
 | url = https://books.google.com/books?id=ruSqmB7LWOcC&pg=PA5
 | year = 2008| volume = 7
 }}</ref> (or the '''relative topology''',<ref name=ttd/> or the '''induced topology''',<ref name=ttd/> or the '''trace topology''').<ref>{{citation
 | last = Pinoli | first = Jean-Charles
 | contribution = The Geometric and Topological Framework
 | date = June 2014
 | doi = 10.1002/9781118984574.ch26
 | isbn = 9781118984574
 | pages = 57–69
 | publisher = Wiley
 | title = Mathematical Foundations of Image Processing and Analysis 2}}; see Section 26.2.4. Submanifolds, p. 59</ref>

== Definition ==

Given a topological space <math>(X, \tau)</math> and a [[subset]] <math>S</math> of <math>X</math>, the '''subspace topology''' on <math>S</math> is defined by
:<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
That is, a subset of <math>S</math> is open in the subspace topology [[if and only if]] it is the [[intersection (set theory)|intersection]] of <math>S</math> with an [[open set]] in <math>(X, \tau)</math>. If <math>S</math> is equipped with the subspace topology then it is a topological space in its own right, and is called a '''subspace''' of <math>(X, \tau)</math>. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset <math>S</math> of <math>X</math> as the [[coarsest topology]] for which the [[inclusion map]]
:<math>\iota: S \hookrightarrow X</math>
is [[continuous (topology)|continuous]].

More generally, suppose  <math>\iota</math> is an [[Injective function|injection]] from a set <math>S</math> to a topological space <math>X</math>. Then the subspace topology on <math>S</math> is defined as the coarsest topology for which <math>\iota</math> is continuous. The open sets in this topology are precisely the ones of the form <math>\iota^{-1}(U)</math> for <math>U</math> open in <math>X</math>. <math>S</math> is then [[homeomorphic]] to its image in <math>X</math> (also with the subspace topology) and <math>\iota</math> is called a [[topological embedding]].

A subspace <math>S</math> is called an '''open subspace''' if the injection <math>\iota</math> is an [[open map]], i.e., if the forward image of an open set of <math>S</math> is open in <math>X</math>. Likewise it is called a '''closed subspace''' if the injection <math>\iota</math> is a [[closed map]].

== Terminology ==

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever <math>S</math> is a subset of <math>X</math>, and <math>(X, \tau)</math> is a topological space, then the unadorned symbols "<math>S</math>" and "<math>X</math>" can often be used to refer both to <math>S</math> and <math>X</math> considered as two subsets of <math>X</math>, and also to <math>(S,\tau_S)</math> and <math>(X,\tau)</math> as the topological spaces, related as discussed above. So phrases such as "<math>S</math> an open subspace of <math>X</math>" are used to mean that <math>(S,\tau_S)</math> is an open subspace of <math>(X,\tau)</math>, in the sense used above; that is: (i) <math>S \in \tau</math>; and (ii) <math>S</math> is considered to be endowed with the subspace topology.

== Examples ==
In the following, <math>\mathbb{R}</math> represents the [[real number]]s with their usual topology.
* The subspace topology of the [[natural number]]s, as a subspace of <math>\mathbb{R}</math>, is the [[discrete topology]].
* The [[rational number]]s <math>\mathbb{Q}</math> considered as a subspace of <math>\mathbb{R}</math> do not have the discrete topology ({0} for example is not an open set in <math>\mathbb{Q}</math> because there is no open subset of <math>\mathbb{R}</math> whose intersection with <math>\mathbb{Q}</math> can result in ''only'' the [[Singleton (mathematics)|singleton]] {0}). If ''a'' and ''b'' are rational, then the intervals (''a'', ''b'') and [''a'', ''b''] are respectively open and closed, but if ''a'' and ''b'' are irrational, then the set of all rational ''x'' with ''a'' < ''x'' < ''b'' is both open and closed.
* The set [0,1] as a subspace of <math>\mathbb{R}</math> is both open and closed, whereas as a subset of <math>\mathbb{R}</math> it is only closed.
* As a subspace of <math>\mathbb{R}</math>, [0, 1] &cup; [2, 3] is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a [[disconnected space]].
* Let ''S'' = [0, 1) be a subspace of the real line <math>\mathbb{R}</math>. Then [0, {{frac|1|2}}) is open in ''S'' but not in <math>\mathbb{R}</math> (as for example the intersection between (-{{frac|1|2}}, {{frac|1|2}}) and ''S'' results in [0, {{frac|1|2}})). Likewise [{{frac|1|2}}, 1) is closed in ''S'' but not in <math>\mathbb{R}</math> (as there is no open subset of <math>\mathbb{R}</math> that can intersect with [0, 1) to result in [{{frac|1|2}}, 1)). ''S'' is both open and closed as a subset of itself but not as a subset of <math>\mathbb{R}</math>.

== Properties ==

The subspace topology has the following characteristic property. Let <math>Y</math> be a subspace of <math>X</math> and let <math>i : Y \to X</math> be the inclusion map. Then for any topological space <math>Z</math> a map <math>f : Z\to Y</math> is continuous [[if and only if]] the composite map <math>i\circ f</math> is continuous. 
[[Image:Subspace-01.svg|center|Characteristic property of the subspace topology]]
This property is characteristic in the sense that it can be used to define the subspace topology on <math>Y</math>.

We list some further properties of the subspace topology. In the following let <math>S</math> be a subspace of <math>X</math>.

* If <math>f:X\to Y</math> is continuous then the restriction to <math>S</math> is continuous.
* If <math>f:X\to Y</math> is continuous then <math>f:X\to f(X)</math> is continuous.
* The closed sets in <math>S</math> are precisely the intersections of <math>S</math> with closed sets in <math>X</math>.
* If <math>A</math> is a subspace of <math>S</math> then <math>A</math> is also a subspace of <math>X</math> with the same topology. In other words, the subspace topology that <math>A</math> inherits from <math>S</math> is the same as the one it inherits from <math>X</math>.
* Suppose <math>S</math> is an open subspace of <math>X</math> (so <math>S\in\tau</math>). Then a subset of <math>S</math> is open in <math>S</math> if and only if it is open in <math>X</math>.
* Suppose <math>S</math> is a closed subspace of <math>X</math> (so <math>X\setminus S\in\tau</math>). Then a subset of <math>S</math> is closed in <math>S</math> if and only if it is closed in <math>X</math>.
* If <math>B</math> is a [[basis (topology)|basis]] for <math>X</math> then <math>B_S = \{U\cap S : U \in B\}</math> is a basis for <math>S</math>.
* The topology induced on a subset of a [[metric space]] by restricting the [[metric (mathematics)|metric]] to this subset coincides with subspace topology for this subset.

== Preservation of topological properties ==

If a topological space having some [[topological property]] implies its subspaces have that property, then we say the property is '''hereditary'''. If only closed subspaces must share the property we call it '''weakly hereditary'''.

* Every open and every closed subspace of a [[completely metrizable]] space is completely metrizable.
* Every open subspace of a [[Baire space]] is a Baire space.
* Every closed subspace of a [[compact space]] is compact.
* Being a [[Hausdorff space]] is hereditary.
* Being a [[normal space]] is weakly hereditary.
* [[Total boundedness]] is hereditary.
* Being [[totally disconnected]] is hereditary.
* [[First countability]] and [[second countability]] are hereditary.

== See also==
* the dual notion [[Quotient space (topology)|quotient space]] 
* [[product topology]]
* [[direct sum topology]]

== Notes ==
{{reflist}}

== References ==
* [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Elements of Mathematics: General Topology'', Addison-Wesley (1966)
* {{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]] | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}}
* Willard, Stephen. ''General Topology'', Dover Publications (2004) {{ISBN|0-486-43479-6}}

[[Category:Topology]]
[[Category:General topology]]