Editing Pointed space (section)

== Operations on pointed spaces ==

* A '''subspace''' of a pointed space <math>X</math> is a [[Subspace (topology)|topological subspace]] <math>A \subseteq X</math> which shares its basepoint with <math>X</math> so that the [[inclusion map]] is basepoint preserving.
* One can form the '''[[Quotient space (topology)|quotient]]''' of a pointed space <math>X</math> under any [[equivalence relation]]. The basepoint of the quotient is the image of the basepoint in <math>X</math> under the quotient map.
* One can form the '''[[Product (category theory)|product]]''' of two pointed spaces <math>\left(X, x_0\right),</math> <math>\left(Y, y_0\right)</math> as the [[Product (topology)|topological product]] <math>X \times Y</math> with <math>\left(x_0, y_0\right)</math>serving as the basepoint.
* The '''[[coproduct]]''' in the category of pointed spaces is the {{em|[[wedge sum]]}}, which can be thought of as the 'one-point union' of spaces.
* The '''[[smash product]]''' of two pointed spaces is essentially the [[Quotient space (topology)|quotient]] of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a [[symmetric monoidal category]] with the pointed [[0-sphere]] as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as [[Compactly generated space|compactly generated]] [[Weak Hausdorff space|weak Hausdorff]] ones.
* The '''[[reduced suspension]]''' <math>\Sigma X</math> of a pointed space <math>X</math> is (up to a [[homeomorphism]]) the smash product of <math>X</math> and the pointed circle <math>S^1.</math>
* The reduced suspension is a functor from the category of pointed spaces to itself. This functor is [[left adjoint]] to the functor <math>\Omega</math> taking a pointed space <math>X</math> to its [[loop space]] <math>\Omega X</math>.