1.2. Topological operations on topological spaces
Problem. Let X =
with the coordinate t (we presuppo
topology on
and let Y be the torus
coordinates (p,ip. We can imagine the torus as the surface of a d
nut. The map X Y is given by the formulas p(t) = at
ijj(t) = /?£ mod 27r. Describe the topology induced on X =
map. How does this topology depend on the numbers a and (3
Quotient topology. Suppose an equivalence relation ~ is giv
a topological space X, i.e., a subset A C X x X is chosen so t
(1) (x, x) e A for all x e X;
(2) {x,y)eA^(y,x)eA;
(3) (x,y),(x,z) e A=* (y,z) e A.
Then a topology, called the quotient topology, on the set of equiv
classes X/~ is defined in the following way. A set U is open
preimage under the canonical projection X X/~ is open.
space X/~ supplied with the quotient topology is called the q
space of X by ~.
Example. X = R1. Suppose we have the following equivalenc
tion: a ~ b == 3A 7^ 0 : a = Xb. Then X/~ consists of two
One of these points is an open set, and the other one is not.
Any subspace Z C X defines an equivalence relation on X:
if either both x and y belong to Z or x y. The correspo
topological quotient space is denoted by X/Z.
Problem. Let X be the space of all real 2 x 2 matrices
The equivalence relation is A ~ B £= A = LBL~l, where L
invertible matrix. Describe the quotient set X/~ and the qu
Definition. A topological space X is path-connected if for an
of its points x, y there is a continuous map / : [0,1] X suc
/(0) = x and /(l ) = y. Maximal path-connected topologica
spaces of a topological space will be called its path-connected c
nents or briefly path-components.
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