JOIN (TOPOLOGY)

In topology, a field of mathematics, the 'join' of two topological spaces ''A'' and ''B'', often denoted by $Astar\; B$, is defined to be the quotient space
:$A\; imes\; B\; imes\; I\; /\; R,\; ,$
where ''I'' is the interval [0, 1] and ''R'' is the relation defined by
:$(a,\; b\_1,\; 0)\; sim\; (a,\; b\_2,\; 0)\; quadmbox\{for\; all\; \}\; a\; in\; A\; mbox\{\; and\; \}\; b\_1,b\_2\; in\; B,$
:$(a\_1,\; b,\; 1)\; sim\; (a\_2,\; b,\; 1)\; quadmbox\{for\; all\; \}\; a\_1,a\_2\; in\; A\; mbox\{\; and\; \}\; b\; in\; B.$
In effect, one is collapsing $A\; imes\; B\; imes\; \{0\}$ to $A$ and $A\; imes\; B\; imes\; \{1\}$ to $B$.
Intuitively, $Astar\; B$ is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in ''A'' to every point in ''B''.

Contents

Examples

See also

References

Examples

★ The join of ''A'' and ''B'', regarded as subsets of ''n''-dimensional Euclidean space is homotopy equivalent to the space of paths in ''n''-dimensional Euclidean space, beginning in ''A'' and ending in ''B''.

★ The join of a space ''X'' with a one-point space is called the cone $Lambda\; X$ of ''X''.

★ The join of a space ''X'' with $S^0$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension $SX$ of ''X''.

★ The join of the spheres $S^n$ and $S^m$ is the sphere $S^\{n+m+1\}$.

See also

★ Cone (topology)

★ Suspension (topology)

References

★ Hatcher, Allen, ''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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