Cone (topology)

In the mathematical sub-area of topology , a cone over a space is a certain set of points constructed from this space, which itself naturally forms a topological space . In the Euclidean case this is actually homeomorphic to a geometric cone , but in general the topological definition is more comprehensive. Mainly, cones are considered over topological spaces in algebraic topology .

definition

Cone over a circle. The original space is colored blue, the collapsed end point is colored green.

Be a topological space. The cone above is defined as the amount ${\displaystyle X}$ ${\displaystyle X}$

${\displaystyle CX:=(X\times [0;1])/(X\times \{1\})}$

provided with the quotient topology with respect to the canonical projection .

The name comes from the Latin word conus for cone. ${\displaystyle C}$

In detail this means:

Let there be a topological space and the real unit interval with the subspace topology . Continue through on the product of these two spaces ${\displaystyle X}$ ${\displaystyle [0;1]\subset \mathbb {R} }$ ${\displaystyle X\times [0;1]}$

${\displaystyle x\sim y:\Leftrightarrow x=y\lor x,y\in (X\times \{1\})}$

explains an equivalence relation . Sit now

${\displaystyle CX=(X\times [0;1])/(X\times \{1\}):=(X\times [0;1])/\sim }$

as the factor space and consider the canonical projection

${\displaystyle p\colon X\times [0;1]\to CX;x\mapsto [x]_{\sim }}$.

A subset should now be called open if and only if its archetype is open in . The system of this open quantities actually forms a topology on the thus created space is the cone over . ${\displaystyle U\subseteq CX}$ ${\displaystyle p^{-1}(U)}$ ${\displaystyle X\times [0;1]}$ ${\displaystyle CX}$ ${\displaystyle X}$

To put it clearly, the top surface of the cylinder is combined into a single point. ${\displaystyle X\times [0;1]}$

properties

• Each topological space can be understood as a subspace of its cone by identifying with . ${\displaystyle X}$ ${\displaystyle X\times \{1\}}$

• The cone of a non-empty space is always contractible by virtue of homotopy . ${\displaystyle h_{t}([x;s]_{\sim })=[(x;(1-t)s)]_{\sim }}$
  • Together with the first property, there is a canonical embedding of any (non-empty) space in a contractible space, which explains the meaning of the cone in algebraic topology.

• The cone over a topological space is homeomorphic to the mapping cone of the identity of this space.

• Each cone is path-connected , i.e. in particular also connected .

• If it can be embedded in a Euclidean space , it is homeomorphic to a geometric cone. ${\displaystyle X}$ ${\displaystyle CX}$
  • Of particular importance in the present case that a cohesive, compact subset of is. (see examples ) ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{2}}$

• If more generally compact and Hausdorffsch , then its cone corresponds to the union of all lines of points to a common point. ${\displaystyle X}$ ${\displaystyle x\in X}$

• Is a CW complex , so too . ${\displaystyle X}$ ${\displaystyle CX}$

Examples

• The cone over a - simplex is a simplex. ${\displaystyle n}$ ${\displaystyle n+1}$
  • For a point there is in particular a line segment , a triangle and a tetrahedron . ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle C\{x\}}$ ${\displaystyle C(C\{x\})}$ ${\displaystyle C(C(C\{x\}))}$
• The cone over a polygon corresponds to the pyramid with a base ${\displaystyle P}$ ${\displaystyle P}$
• The topological cone over a filled circle is the classic circular cone. (see picture )
• The topological cone over a circular line is the lateral surface of a circular cone; this, in turn, is topologically equivalent to the full circle, in that the point is clearly depressed.
  • In general, homeomorphism applies . ${\displaystyle CS^{n-1}\cong D^{n}}$

Reduced cone

Let now a dotted area , so is reduced cone over defined as ${\displaystyle (X;x_{0})}$ ${\displaystyle X}$

${\displaystyle C_{*}X=X\times [0,1]/(X\times \left\{0\right\})\cup (\left\{x_{0}\right\}\times [0,1])}$

with the quotient topology.

With as a base point itself becomes a dotted space again and the above-mentioned inclusion becomes a base point-preserving embedding. ${\displaystyle [(x_{0};0)]_{\sim }}$ ${\displaystyle C_{*}X}$ ${\displaystyle x\mapsto [(x;1)]_{\sim }}$

The reduced cone is equal to the reduced image cone of the identity.

Cone functor

In category theory , the assignment induces an endofunctor on the category of topological spaces. ${\displaystyle X\mapsto CX}$ ${\displaystyle C:\mathbf {Top} \to \mathbf {Top} }$ ${\displaystyle \mathbf {Top} }$

This also assigns the mapping to each continuous mapping that is explained by. ${\displaystyle f\in \operatorname {Mor} (X;Y)}$ ${\displaystyle C(f)\in \operatorname {Mor} (CX;CY)}$ ${\displaystyle [(x;t)]_{\sim _{CX}}\mapsto [(f(x);t)]_{\sim _{CY}}}$

The same applies to in the category of dotted topological spaces. ${\displaystyle X\mapsto C_{*}X}$ ${\displaystyle \mathbf {Top*} }$

Note: The notation used here should not be confused with the mapping cone for continuous or the function space of all continuous mappings on a topological space . ${\displaystyle Cf}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {C}}(X)}$ ${\displaystyle X}$

See also

• Suspension

Individual evidence

1. ↑ Allen Hatcher: Algebraic topology. 9, Cambridge University Press, Cambridge 2002, quoted in: math.cornell.edu.Retrieved July 2, 2012.
2. ^ Klaus Jänich: Topology. 8th edition, 51f., Springer, Berlin 2008.
3. ↑ Lothar Tschampel: Topology 2: References to Algebra. Buch-MAT 3.B, 1st edition, Buch-X-Verlag, Berlin 2011.
4. ^ Roman Goebel: Continuity of the cone functor. in: Topology and its applications. 132, pp. 235–250, 2003, quoted from: wiki.helsinki.fi.Retrieved July 4, 2012.