1.2. Topological operations on topological spaces

Problem. Let X =

R1

with the coordinate t (we presuppo

topology on

R1),

and let Y be the torus

T2

=

S1

x

S1

with

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 =

R1

b

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

topology.

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.

Problem. Let X =

R1

with the coordinate t (we presuppo

topology on

R1),

and let Y be the torus

T2

=

S1

x

S1

with

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 =

R1

b

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

topology.

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.