Subspace topology

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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 natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology).

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Definition

Given a topological space <math>(X, \tau)</math> and a subset <math>S\sube 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 iff it is the 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.

If <math>S</math> is open, closed or dense in <math>(X, \tau)</math> we call <math>(S, \tau_S)</math> an open subspace, closed subspace or dense subspace of <math>(X, \tau)</math>.

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.

More generally, suppose <math>i : S \to X</math> is an 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

  • Given the real numbers with the usual topology the subspace topology of the natural numbers, as a subspace of the real numbers, is the discrete topology.
  • The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 is not open in Q).
  • Let S = [0,1) be a subspace the real line R. Then [0,½) 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

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 iff the composite map <math>i\circ f</math> is continuous. Image:Subspace-01.png 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 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>. Then a subspace of <math>S</math> is open in <math>S</math> iff it is open in <math>X</math>.
  • 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> iff it is closed in <math>X</math>.
  • If <math>B</math> is a base 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 to this subset coincides with subspace topology for this subset.
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Preservation of topological properties

If a topological space has a certain topological property and every subspace shares the same property we say the topological property is hereditary. If only closed subspaces share the property we call it weakly hereditary.

  • every open subspace of a Baire space is a Baire space
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References

  • Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
  • Steen, Lynn A. and Seeback, J. Arthur Jr., Counterexamples in Topology, Holt, Rinehart and Winston (1970) ISBN 0030794854.
  • Wilard, Stephen. General Topology, Dover Publications (2004) ISBN 0486434796
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See also

it:Topologia del sottoinsieme nl:Deelruimtetopologie pl:Podprzestrzeń (topologia)

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