Cone (topology) · Lablypedia

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Definitions

Formally, the cone of X is defined as:

C X = (X \times [0, 1]) \cup_{p} v = \lim_{\to} ((X \times [0, 1]) ↩ (X \times {0}) \overset{p}{\to} v),

where $v$ is a point (called the vertex of the cone) and $p$ is the projection to that point. In other words, it is the result of attaching the cylinder $X \times [0, 1]$ by its face $X \times {0}$ to a point $v$ along the projection $p : (X \times {0}) \to v$ .

If $X$ is a non-empty compact subspace of Euclidean space, the cone on $X$ is homeomorphic to the union of segments from $X$ to any fixed point $v \notin X$ such that these segments intersect only in $v$ 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.

The cone is a special case of a join: $C X ≃ X ⋆ {v} =$ the join of $X$ with a single point $v \notin X$ .

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Examples

Here we often use a geometric cone ( $C X$ where $X$ is a non-empty compact subspace of Euclidean space). The considered spaces are compact, so we get the same result up to homeomorphism.

The cone over a point p of the real line is a line-segment in $R^{2}$ , ${p} \times [0, 1]$ .
The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example).
The cone over a polygon P is a pyramid with base P.
The cone over a disk is the solid cone of classical geometry (hence the concept's name).
The cone over a circle given by

{(x, y, z) \in R^{3} ∣ x^{2} + y^{2} = 1 and z = 0}

is the curved surface of the solid cone:

{(x, y, z) \in R^{3} ∣ x^{2} + y^{2} = (z - 1)^{2} and 0 \leq z \leq 1} .

This in turn is homeomorphic to the closed disc.

More general examples:

The cone over an n-sphere is homeomorphic to the closed (n + 1)-ball.
The cone over an n-ball is also homeomorphic to the closed (n + 1)-ball.
The cone over an n-simplex is an (n + 1)-simplex.

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

h_{t} (x, s) = (x, (1 - t) s)

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 $C X$ 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 $C X$ will be finer than the set of lines joining X to a point.

Cone functor

The map $X \mapsto C X$ induces a functor $C : T o p \to T o p$ on the category of topological spaces Top. If $f : X \to Y$ is a continuous map, then $C f : C X \to C Y$ is defined by

(C f) ([x, t]) = [f (x), t]

where square brackets denote equivalence classes.

Reduced cone

If $(X, x_{0})$ is a pointed space, there is a related construction, the reduced cone, given by

(X \times [0, 1]) / (X \times {0} \cup {x_{0}} \times [0, 1])

where we take the basepoint of the reduced cone to be the equivalence class of $(x_{0}, 0)$ . With this definition, the natural inclusion $x \mapsto (x, 1)$ becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.