2

Here are some examples of topological spaces.

Example, r = 2

X

, i.e., we consider any subset in X as an ope

This is equivalent to declare each point x G l a n open subset.

topology is called discrete.

Sometimes it is convenient to introduce a topology r by

of a basis of the topology. A basis of a topology r — { Va } is a

{ W\ } C r such that each open set can be represented as the

of a (probably infinite) family of sets from W\.

Recall that a metric space is a set M endowed with a real-v

function p: M x M — R+ possessing the following properties:

(1) p{x,y) = 0 = x^y]

(2) p(x,y) = p(y,x)]

(3) p(x,y) + p(y,z) ^ p(x,z).

Example (topology of a metric space). We take the set of all

balls

V£,x = {yeM\ \x-y\ e}

as the basis of the topology of a metric space.

The spaces R1 and Rn are metric spaces. The standard topo

in these spaces are defined as the topologies of the correspo

metric spaces.

It is possible to choose another basis in the topology of R n .

basis consists of open balls with rational radii and rational c

nates of the centers. This basis determines the same topology,

contrast to the above basis this one is countable.

Exercise. Consider the basis consisting of parallelepipeds in Rn

edges parallel to the coordinate axes. Is the topology it determ

different one?

Example (an exotic topology on R1). For the open sets in

take those sets that are open in the usual sense, and periodic

period 1 (i.e., t G U == t + 1 6 U for each open set U).

Definition. A subset A C X is called closed if its complement

is open. The closure A of a subset A C X is the minimal clos

containing A.

Here are some examples of topological spaces.

Example, r = 2

X

, i.e., we consider any subset in X as an ope

This is equivalent to declare each point x G l a n open subset.

topology is called discrete.

Sometimes it is convenient to introduce a topology r by

of a basis of the topology. A basis of a topology r — { Va } is a

{ W\ } C r such that each open set can be represented as the

of a (probably infinite) family of sets from W\.

Recall that a metric space is a set M endowed with a real-v

function p: M x M — R+ possessing the following properties:

(1) p{x,y) = 0 = x^y]

(2) p(x,y) = p(y,x)]

(3) p(x,y) + p(y,z) ^ p(x,z).

Example (topology of a metric space). We take the set of all

balls

V£,x = {yeM\ \x-y\ e}

as the basis of the topology of a metric space.

The spaces R1 and Rn are metric spaces. The standard topo

in these spaces are defined as the topologies of the correspo

metric spaces.

It is possible to choose another basis in the topology of R n .

basis consists of open balls with rational radii and rational c

nates of the centers. This basis determines the same topology,

contrast to the above basis this one is countable.

Exercise. Consider the basis consisting of parallelepipeds in Rn

edges parallel to the coordinate axes. Is the topology it determ

different one?

Example (an exotic topology on R1). For the open sets in

take those sets that are open in the usual sense, and periodic

period 1 (i.e., t G U == t + 1 6 U for each open set U).

Definition. A subset A C X is called closed if its complement

is open. The closure A of a subset A C X is the minimal clos

containing A.