Suspension (topology)

In topology, the suspension SX of a topological space X is the quotient space:

of the product of X with the unit interval I = [0, 1]. Thus, X is stretched into a cylinder and then both ends are collapsed to points. One views X as "suspended" between the end points. One can also view the suspension as two cones on X glued together at their base (or as a quotient of a single cone).

Given a continuous map ${\displaystyle f:X\rightarrow Y,}$ there is a map ${\displaystyle Sf:SX\rightarrow SY}$ defined by ${\displaystyle Sf([x,t]):=[f(x),t].}$ This makes ${\displaystyle S}$ into a functor from the category of topological spaces into itself. In rough terms S increases the dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.

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