Cone (topology)

In topology, especially algebraic topology, the cone formula_1 of a topological space formula_2 is the quotient space:
of the product of "X" with the unit interval formula_4. Intuitively, this construction makes "X" into a cylinder and collapses one end of the cylinder to a point.
If formula_2 is a compact subspace of Euclidean space, the cone on formula_2 is homeomorphic to the union of segments from formula_2 to any fixed point formula_8 such that these segments intersect only by formula_9 itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.
Examples.
Here we often use geometric cone (defined in the introduction) instead of the topological one. The considered spaces are compact, so we get the same result up to homeomorphism.
Properties.
All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy
The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.
When "X" is compact and Hausdorff (essentially, when "X" can be embedded in Euclidean space), then the cone formula_1 can be visualized as the collection of lines joining every point of "X" to a single point. However, this picture fails when "X" is not compact or not Hausdorff, as generally the quotient topology on formula_1 will be finer than the set of lines joining "X" to a point.
Cone functor.
The map formula_16 induces a functor formula_17 on the category of topological spaces Top. If formula_18 is a continuous map, then formula_19 is defined by 
where square brackets denote equivalence classes.
Reduced cone.
If formula_21 is a pointed space, there is a related construction, the reduced cone, given by 
where we take the basepoint of the reduced cone to be the equivalence class of formula_23. With this definition, the natural inclusion formula_24 becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.