CONE (TOPOLOGY)

In topology, especially algebraic topology, the 'cone' ''CX'' 'of a topological space' ''X'' is the quotient space:
:$CX\; =\; (X\; imes\; I)/(X\; imes\; \{0\}),$
of the product of ''X'' with the unit interval ''I'' = [0, 1].
Intuitively we make ''X'' into a cylinder and collapse one end of the cylinder to a point.
If ''X'' sits inside Euclidean space, the cone on ''X'' is homeomorphic to the union of lines from ''X'' to another point. That is, the topological cone agrees with the geometric cone when defined. However, the topological cone construction is more general.

Contents

Examples

Properties

Reduced cone

Cone functor

See also

References

Examples

★ The cone over a point ''p'' of the real line is the interval {''p''} x [0,1].

★ The cone over two points {0,1} is a "V" shape with endpoints at {0} and {1}.

★ The cone over an interval ''I'' of the real line is a filled-in triangle, 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 is the curved surface of the solid cone:
::$\{(x,y,z)\; in\; mathbb\; R^3\; mid\; x^2\; +\; y^2\; =\; z^2\; mbox\{\; and\; \}\; 0leq\; zleq\; 1\}.$
:This in turn is homeomorphic to the closed disc.

★ In general, the cone over an n-sphere is 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 ''CX'' 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 ''CX'' will be finer than the set of lines joining ''X'' top a point.

Reduced cone

If $(X,x\_0)$ is a pointed space, there is a related construction, the 'reduced cone', given by
:$X\; imes\; [0,1]\; /\; (X\; imes\; left\{0$
ight})
cup(left{x_0
ight} imes [0,1])
With this definition, the natural inclusion $xmapsto\; (x,1)$ becomes a based map, where we take $(x\_0,0)$ to be the basepoint of the reduced cone.

Cone functor

The map $Xmapsto\; CX$ induces a functor on the category of topological spaces 'Top'.

See also

★ Cone

★ Suspension (topology)

★ Mapping cone

★ Join (topology)

References

★ Allen Hatcher, ''Algebraic topology.'' Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0

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