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| In [[topology]] and related areas of [[mathematics]], a '''subspace''' of a [[topological space]] ''X'' is a [[subset]] ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the '''subspace topology''' (or the '''relative topology''', or the '''induced topology''', or the '''trace topology''').
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| == Definition ==
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| 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
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| :<math>\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.</math>
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| 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.
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| 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]]
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| :<math>\iota: S \hookrightarrow X</math>
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| is [[continuous (topology)|continuous]].
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| More generally, suppose <math>i</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>i</math> is continuous. The open sets in this topology are precisely the ones of the form <math>i^{-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>i</math> is called a [[topological embedding]].
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| == Examples ==
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| In the following, '''R''' represents the [[real number]]s with their usual topology.
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| * The subspace topology of the [[natural number]]s, as a subspace of '''R''', is the [[discrete topology]].
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| * The [[rational number]]s '''Q''' considered as a subspace of '''R''' do not have the discrete topology (the point 0 for example is not an open set in '''Q'''). 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 ''x'' with <math>a<x<b</math> is both open and closed.
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| * The set [0,1] as a subspace of '''R''' is both open and closed, whereas as a subset of '''R''' it is only closed.
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| * As a subspace of '''R''', <math>[0,1]\cup[2,3]</math> is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a [[disconnected space]].
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| * Let ''S'' = [0,c) be a subspace of the real line '''R'''. Then [0,c/2) is open in ''S'' but not in '''R'''. Likewise [½, 1) is closed in ''S'' but not in '''R'''. ''S'' is both open and closed as a subset of itself but not as a subset of '''R'''.
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| == Properties ==
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| 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.
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| [[Image:Subspace-01.png|center|Characteristic property of the subspace topology]]
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| This property is characteristic in the sense that it can be used to define the subspace topology on <math>Y</math>.
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| We list some further properties of the subspace topology. In the following let <math>S</math> be a subspace of <math>X</math>.
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| * If <math>f:X\to Y</math> is continuous the restriction to <math>S</math> is continuous.
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| * If <math>f:X\to Y</math> is continuous then <math>f:X\to f(X)</math> is continuous.
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| * The closed sets in <math>S</math> are precisely the intersections of <math>S</math> with closed sets in <math>X</math>.
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| * 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>.
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| * Suppose <math>S</math> is an open subspace of <math>X</math>. Then a subspace of <math>S</math> is open in <math>S</math> if and only if it is open in <math>X</math>.
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| * Suppose <math>S</math> is a closed subspace of <math>X</math>. Then a subspace of <math>S</math> is closed in <math>S</math> if and only if it is closed in <math>X</math>.
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| * If <math>B</math> is a [[basis (topology)|base]] for <math>X</math> then <math>B_S = \{U\cap S : U \in B\}</math> is a basis for <math>S</math>.
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| * 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.
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| == Preservation of topological properties ==
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| 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'''.
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| * Every open and every closed subspace of a [[completely metrizable]] space is completely metrizable.
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| * Every open subspace of a [[Baire space]] is a Baire space.
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| * Every closed subspace of a [[compact space]] is compact.
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| * Being a [[Hausdorff space]] is hereditary.
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| * Being a [[normal space]] is weakly hereditary.
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| * [[Total boundedness]] is hereditary.
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| * Being [[totally disconnected]] is hereditary.
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| * [[First countability]] and [[second countability]] are hereditary.
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| == See also==
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| * the dual notion [[quotient space]]
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| * [[product topology]]
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| * [[direct sum topology]]
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| == References ==
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| * Bourbaki, Nicolas, ''Elements of Mathematics: General Topology'', Addison-Wesley (1966)
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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|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995}}
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| * Willard, Stephen. ''General Topology'', Dover Publications (2004) ISBN 0-486-43479-6
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| [[Category:Topology]]
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| [[Category:General topology]]
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The writer is known as Irwin. California is where I've usually been living and I love every working day residing right here. Hiring is her working day job now and she will not alter it anytime soon. To collect badges is what her family and her appreciate.
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