1.1. Topological spaces and homeomorphisms

Definition. A map / : X — Y of a topological space J t o a t

ical space Y is called continuous with respect to the given topo

if the preimage of any open set in Y is open in X.

Problem. Verify that for the spaces X = Y —

R1

endowed w

standard topology this definition coincides with the standard

definition from Calculus.

Any map / : X — • Y of a space X with discrete topology

tinuous.

Induced topology. Let Y be a topological space, and let X

be a subset. Then one can introduce a topology on X by consi

the intersections of X with open subsets in Y as open subsets

This topology is said to be the induced topology.

In a more general setting, the induced topology is defin

follows. Let Y be a topological space, and let / : X — » Y be

The induced topology in X consists of all preimages of open sets

(The map / is then continuous with respect to the induced topo

For the inclusion map / of a subset I c 7 w e recover the pr

definition.

ABOUT THE FIGURES.

Whenever we draw a figure, e.g. a

curve, or a surface in space, we mean implicitly that the topolo

this curve or surface is induced from the standard topology

ambient space

R2

or

R3.

Definition. A map / : X — Y is called a homeomorphism

continuous and has an inverse map /

_ 1

: Y — X, f o

jf-1

— i

is also continuous. If a homeomorphism X —» Y exists, th

spaces X and Y are called homeomorphic. We write this relat

the following way: X « Y.

Topology studies topological spaces and continuous maps c

ered up to homeomorphisms.

Example. The boundary of the square, and the circle are h

morphic (see Figure 1: the homeomorphism takes the point A

point B).

Definition. A map / : X — Y of a topological space J t o a t

ical space Y is called continuous with respect to the given topo

if the preimage of any open set in Y is open in X.

Problem. Verify that for the spaces X = Y —

R1

endowed w

standard topology this definition coincides with the standard

definition from Calculus.

Any map / : X — • Y of a space X with discrete topology

tinuous.

Induced topology. Let Y be a topological space, and let X

be a subset. Then one can introduce a topology on X by consi

the intersections of X with open subsets in Y as open subsets

This topology is said to be the induced topology.

In a more general setting, the induced topology is defin

follows. Let Y be a topological space, and let / : X — » Y be

The induced topology in X consists of all preimages of open sets

(The map / is then continuous with respect to the induced topo

For the inclusion map / of a subset I c 7 w e recover the pr

definition.

ABOUT THE FIGURES.

Whenever we draw a figure, e.g. a

curve, or a surface in space, we mean implicitly that the topolo

this curve or surface is induced from the standard topology

ambient space

R2

or

R3.

Definition. A map / : X — Y is called a homeomorphism

continuous and has an inverse map /

_ 1

: Y — X, f o

jf-1

— i

is also continuous. If a homeomorphism X —» Y exists, th

spaces X and Y are called homeomorphic. We write this relat

the following way: X « Y.

Topology studies topological spaces and continuous maps c

ered up to homeomorphisms.

Example. The boundary of the square, and the circle are h

morphic (see Figure 1: the homeomorphism takes the point A

point B).