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
Be a topological space. The cone above is defined as the amount
provided with the quotient topology with respect to the canonical projection .
The name comes from the Latin word conus for cone.
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
explains an equivalence relation . Sit now
as the factor space and consider the canonical projection
- .
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 .
To put it clearly, the top surface of the cylinder is combined into a single point.
properties
- Each topological space can be understood as a subspace of its cone by identifying with .
- The cone of a non-empty space is always contractible by virtue of homotopy .
- 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.
- Of particular importance in the present case that a cohesive,
- If more generally compact and Hausdorffsch , then its cone corresponds to the union of all lines of points to a common point.
- Is a CW complex , so too .
Examples
- The cone over a - simplex is a simplex.
- For a point there is in particular a
- In general, homeomorphism applies .
Reduced cone
Let now a dotted area , so is reduced cone over defined as
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.
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.
This also assigns the mapping to each continuous mapping that is explained by.
The same applies to in the category of dotted topological spaces.
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 .
See also
Individual evidence
- ↑ Allen Hatcher: Algebraic topology. 9, Cambridge University Press, Cambridge 2002, quoted in: math.cornell.edu.Retrieved July 2, 2012.
- ^ Klaus Jänich: Topology. 8th edition, 51f., Springer, Berlin 2008.
- ↑ Lothar Tschampel: Topology 2: References to Algebra. Buch-MAT 3.B, 1st edition, Buch-X-Verlag, Berlin 2011.
- ^ 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.