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L asked in 科學數學 · 1 decade ago

# 一個拓樸空間的 Suspension 是什麼意思 @@?

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

SX = (X x I) / {(x_1,0) ~ (x_2,0) and (x_1,1) ~ (x_2,1) for all x_1 , x_2 in X} (?)

of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two 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).

http://en.wikipedia.org/wiki/Suspension_%28topolog...

Update:

SX 的表示法我看不太懂

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Let Y be a topological space. An equivalence relation ~ on Y is a relation on Y that is reflexive, symmetric, and transitive. The equivalence class of a point y ∈ Y is the set [y] = {z ∈ Y: z ~ y}. The collection of equivalence classes partitions Y, since ∪[y] = Y and for equivalence classes [y] and [z] either [y] = [z] holds or [y] and [z] are disjoint.The quotient space Y/~ = {[y] : y ∈ Y} is the topological space where the points are the equivalence classes of Y and a subset of Y is considered to be open if the union of its points (which are subsets of Y) is open in Y.The suspension SX of a topological space X is the quotient space X×[0,1]/~ where ~ is the equivalence relation defined as(x1,0) ~ (x2,0) for all x1, x2 ∈ X(x1,1) ~ (x2,1) for all x1, x2 ∈ X(x,t) ~ (x,t) for all (x,t) ∈ X×[0,1].In other words, the points in SX (which are equivalence classes) areX×{0}X×{1}{(x,t)} where (x,t) ∈ X×(0,1).A subset of SX is said to be open if the union of its points, treated as subsets of X×(0,1), is open in the topology of X×[0,1].Intuitively, if we visualize X as a disk, then X×[0,1] would be a cylinder. The equivalence relation would then squeeze the disks X×{0} and X×{1} into points (the green points in the figure), and we can visualize the rest of the cylinder as tapering to green points from the center, since size/radius is irrelevant from a topological point of view.

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