Editing Subspace topology (section)

== 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.