Pointed Space - Operations On Pointed Spaces

Operations On Pointed Spaces

• A subspace of a pointed space X is a topological subspace A ⊆ X which shares its basepoint with X so that the inclusion map is basepoint preserving.
• One can form the quotient of a pointed space X under any equivalence relation. The basepoint of the quotient is the image of the basepoint in X under the quotient map.
• One can form the product of two pointed spaces (X, x₀), (Y, y₀) as the topological product X × Y with (x₀, y₀) serving as the basepoint.
• The coproduct in the category of pointed spaces is the 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 of the direct product and the wedge sum. The smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object.
• The reduced suspension ΣX of a pointed space X is (up to a homeomorphism) the smash product of X and the pointed circle S1.
• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is a left adjoint to the functor taking a based space to its loop space .