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 .
Read more about this topic: Pointed Space
Famous quotes containing the words operations, pointed and/or spaces:
“It may seem strange that any road through such a wilderness should be passable, even in winter, when the snow is three or four feet deep, but at that season, wherever lumbering operations are actively carried on, teams are continually passing on the single track, and it becomes as smooth almost as a railway.”
—Henry David Thoreau (1817–1862)
““Courage!” he said, and pointed toward the land,
“This mounting wave will roll us shoreward soon.”
In the afternoon they came unto a land
In which it seemed always afternoon.”
—Alfred Tennyson (1809–1892)
“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
—Jean Baudrillard (b. 1929)