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 5° - {-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.

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 5° - {-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.