In
topology, especially
algebraic topology, the coneof a
topological space is intuitively obtained by stretching X into a
cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by or by .
Definitions
Formally, the cone of X is defined as:
where is a
point (called the vertex of the cone) and is the
projection to that point. In other words, it is the result of
attaching the
cylinder by its face to a point along the projection .
If is a non-empty
compact subspace of
Euclidean space, the cone on is
homeomorphic to the
union of segments from to any fixed point such that these segments intersect only in 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: the join of with a single point .[1]: 76
Examples
Here we often use a
geometric cone ( where 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 , .
The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
The cone over a
closed intervalI 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
polygonP 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 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
.
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 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 will be
finer than the set of lines joining X to a point.
If 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 . With this definition, the natural inclusion becomes a based map. This construction also gives a functor, from the
category of pointed spaces to itself.