Network (topology)

In mathematics is composite (ger .: join ) topological spaces one on John Milnor declining construction of the topology .

construction

The combination of two intervals (blue and green) is a 3-dimensional polytope (gray).

Association of two topological spaces

Let and be two topological spaces. Their network is defined as follows. The elements of are the couple ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X=X_{1}*X_{2}}$ ${\displaystyle X}$

${\displaystyle (t_{1}x_{1},t_{2}x_{2})}$with , ${\displaystyle x_{1}\in X_{1},x_{2}\in X_{2},t_{1},t_{2}\in \left[0,1\right],t_{1}+t_{2}=1}$

where is an abbreviation for the couple and for all and all ${\displaystyle t_{i}x_{i}}$ ${\displaystyle (t_{i},x_{i})}$ ${\displaystyle x_{1},x_{1}^{\prime }\in X_{1}}$ ${\displaystyle x_{2},x_{2}^{\prime }\in X_{2}}$

${\displaystyle (0x_{1},1x_{2})=(0x_{1}^{\prime },1x_{2})}$ and ${\displaystyle (1x_{1},0x_{2})=(1x_{1},0x_{2}^{\prime })}$

is set. (So ​​all points from are clearly connected to all points from by stretching the length .) ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle 1}$

The topology on is by definition the coarsest topology (the topology with the fewest open sets) with respect to all coordinate mappings ${\displaystyle X}$

${\displaystyle t_{i}\colon X\to \left[0,1\right]}$

${\displaystyle (t_{1}x_{1},t_{2}x_{2})\to t_{i}\quad (i=1,2)}$

and

${\displaystyle x_{i}\colon \left\{(t_{1}x_{1},t_{2}x_{2})\colon t_{i}\not =0\right\}\to X_{i}}$

${\displaystyle (t_{1}x_{1},t_{2}x_{2})\to x_{i}\quad (i=1,2)}$

are steady.

Examples

• The compound of a room with a point is the cone above . ${\displaystyle X}$ ${\displaystyle CX}$ ${\displaystyle X}$
• The connection of a room with the 2-element room is the suspension of . ${\displaystyle X}$ ${\displaystyle S^{0}}$ ${\displaystyle SX}$ ${\displaystyle X}$
• The union of two spheres and is the -dimensional sphere . ${\displaystyle S^{k}}$ ${\displaystyle S^{l}}$ ${\displaystyle (k+l+1)}$ ${\displaystyle S^{k+l+1}}$
• The compound of circles is the -dimensional sphere . ${\displaystyle k}$ ${\displaystyle S^{1}*\ldots *S^{1}}$ ${\displaystyle (2k-1)}$ ${\displaystyle S^{2k-1}}$
• For the Cartesian product of two CAT (0) -spaces and their geodetic boundaries holds . ${\displaystyle X_{1}\times X_{2}}$ ${\displaystyle X_{1},X_{2}}$ ${\displaystyle \partial _{\infty }(X_{1}\times X_{2})=\partial _{\infty }X_{1}*\partial _{\infty }X_{2}}$

Spherical compound

On the connection of two metric spaces and one can define a metric as follows: The distance is the number in the interval for which ${\displaystyle (X_{1},d_{1})}$ ${\displaystyle (X_{2},d_{2})}$ ${\displaystyle d((t_{1}x_{1},t_{2}x_{2}),(s_{1}y_{1},s_{2}y_{2}))}$ ${\displaystyle \left[0,\pi \right]}$

${\displaystyle \cos(d((t_{1}x_{1},t_{2}x_{2}),(s_{1}y_{1},s_{2}y_{2})))={\sqrt {t_{1}s_{1}}}\cos(min\left\{\pi ,d_{1}(x_{1},y_{1})\right\})+{\sqrt {t_{2}s_{2}}}\cos(min\left\{\pi ,d_{2}(x_{2},y_{2})\right\})}$

applies. Note that the constraints on this metric are based on and not the original metrics , but rather give. ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle d_{i}(x,y)}$ ${\displaystyle min\left\{\pi ,d_{i}(x,y)\right\}}$

The metric space is called the spherical compound of the metric spaces and . ${\displaystyle (X_{1}*X_{2},d)}$ ${\displaystyle (X_{1},d_{1})}$ ${\displaystyle (X_{2},d_{2})}$

Association of an infinite number of topological spaces

Let it be a family of topological spaces. The elements of the compound are the tuples ${\displaystyle \left\{X_{j}\colon j\in J\right\}}$ ${\displaystyle X=*_{j\in J}X_{j}}$ ${\displaystyle J}$

${\displaystyle (t_{j}x_{j}\colon j\in J)}$with almost everyone . ${\displaystyle t_{j}\in \left[0,1\right],x_{j}\in X_{j},\sum _{j\in J}t_{j}=1,}$ ${\displaystyle t_{j}=0}$

Two tuples and define the same element if and only if: ${\displaystyle (t_{j}x_{j})}$ ${\displaystyle (u_{j}y_{j})}$

• For everyone is . ${\displaystyle j\in J}$ ${\displaystyle t_{j}=u_{j}}$
• For all true: . ${\displaystyle j\in J}$ ${\displaystyle t_{j}\not =0\Longrightarrow x_{j}=y_{j}}$

The topology on is the coarsest topology (the topology with the fewest open sets) with respect to all coordinate mappings ${\displaystyle X}$

${\displaystyle t_{j}\colon X\to \left[0,1\right]}$

${\displaystyle (t_{i}x_{i})\to t_{j}\quad (j\in J)}$

and

${\displaystyle x_{j}\colon \left\{(t_{i}x_{i})\colon t_{j}\not =0\right\}\to X_{j}}$

${\displaystyle (t_{i}x_{i})\to x_{j}\quad (j\in J)}$

are steady.

Examples

• For a topological group , the countably infinite union is the so-called Milnor space , it is the classifying space for - principal bundles . ${\displaystyle G}$ ${\displaystyle G*G*\ldots *G*\ldots }$ ${\displaystyle G}$

literature

• Tammo tom Dieck: Topology. de Gruyter textbook. Walter de Gruyter & Co., Berlin 1991, ISBN 3-11-013187-0 ; 3-11-012463-7
• Martin R. Bridson; André Haefliger: Metric spaces of non-positive curvature. Basic teaching of the mathematical sciences, 319. Springer, Berlin 1999, ISBN 3-540-64324-9

Individual evidence

1. ↑ Berestovskiĭ, VN: Borsuk's problem on metrization of a polyhedron. (Russian) Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 273-277.