6
1. Topological s
Figure 3
Problem. It is obvious that belonging to the same path-conn
component of X is an equivalence relation. Is the corresponding
tient topology always discrete?
Suspension. Let X be a topological space. The suspension
defined as the quotient space X x J/~, where / is the segment [
and the equivalence relation is the following:
for A ^ - 1 , 1 (x, A) ~ (y, //) =• x = y, A = /x;
(#, 1) ~ (y, 1) for all x,y e X;
(x, -1) ~ (y, -1) for all x,y e X.
In other words, we take the cylinder X xl and contract its
base X x {1} to a point and its lower base X x {—1} to a poin
Figure 3).
Example. The n-dimensional sphere Sn is the topological
homeomorphic to the subset in R n + 1 specified by the eq
x\-\ h#n+i = 1. It can be proved that Y,Sn « Sn+1 (this stat
holds for - {-1,1} as well).
Attaching a topological space by a map. Let X and Y be d
topological spaces, let A C X and let p: A Y be a continuo
Then the operation of attaching X to Y by the map tp produc
space
I U ^ F = ( I U Y)/{a - ip(a)}\
this means that a point a G A is equivalent to the point p(a)
any point not in A and not in the image of ip is equivalent o
itself.
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