Join (topology)

In Topology , a field of Mathematics , the join of two Topological Space s ''A'' and ''B'', often denoted by

A\star B

, is defined to be the Quotient Space
:

A 	imes B 	imes I / R, \,

where ''I'' is the Interval 1 and ''R'' is the relation defined by
:

(a, b_1, 0) \sim (a, b_2, 0) \quad\mbox{for all } a \in A \mbox{ and } b_1,b_2 \in B,

(a_1, b, 1) \sim (a_2, b, 1) \quad\mbox{for all } a_1,a_2 \in A \mbox{ and } b \in B.

In effect, one is collapsing

A	imes B	imes \{0\}

A

and

A	imes B	imes \{1\}

B

.

Intuitively,

A\star 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''.

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''.

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