Join of topological spaces
This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
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Given two topological spaces and , the join of and , denoted , is defined as follows: it is the quotient of the space under the identifications:
Pictorially, we can think of this as the space of all line segments joining points in and , with two line segments meeting only at common endpoints.
Further information: Cone space
The cone space of a topological space can be viewed as the join of with a one-point space.
Further information: suspension
The suspension of a topological space can be viewed as the join of with a two-point space.
The -simplex can be viewed, at least topologically, as the join of one-point spaces.
There is a canonical isomorphism between and , sending to (y,x,1-t)</math>.
There is a canonical isomorphism between and .