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).
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