Download Continuous functions( (الدوال المستمرة introduction The concept of

Lecture 4 : Topology
Lecture : K . Al-Ghurabi
Continuous functions(‫(الدوال المستمرة‬
introduction
The concept of continuous function is basic to much of mathematics
.continuous functions on the real line appear in the first of any calculus book
,and continuous functions in the plane and in a space follow and far behind
,more general kinds of continuous functions arise as one goes further in
mathematics ,in this lecture ,we shall formulate a definition of continuity that
will include all these as special cases ,and we will study various properties of
continuous function ,many of these properties are studied.
Now let X and Y are two topological spaces . A function f :X→ Y is said to be
continuous function if the inverse image of each open set in Y is an open set in
X
Recall that the inverse image of this open set is the set of all points x in X
for which f(x) is belong in this open set and it is empty if this open set dose not
intersect the image set f(x) of f .
Continuity of a function depends not only up on the function f itself ,but
also on the topologies specified for it is domain and range .if we wish emphasis
this fact ,we can Say that f is continuous relative to specific topologies on X and
Y.
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Lecture 4 : Topology
Lecture : K . Al-Ghurabi
Let us that if the topology of the range space Y is given by a basis β ,then to
prove continuity of f it suffices to show that the inverse image of every basis
element is open .the arbitrary open set V of a topological space Y can be
written as a union of basis element .
Definition :
A function f:
is said to be open if the image of any open
set in X is open in Y.
Definition :
A function f:
is said to be closed if the image of any closed
set in X is closed in Y.
Definition :
A function f:
is said to be homeomorphism iff :
1- f is bijective ( 1-1 and onto ).
2- f is continuous .
3is continuous .
Definition :
A space X is said to be homeomorphic to another space Y if There is
a homeomorphism of X onto Y .
 If
is homeomorphism to
we write
.
Definition :
A property of a topological Space X is said to be a topological
property if each homeomorphism of X has that property whenever X
has that property .
Theorem :
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Lecture 4 : Topology
let f:
Lecture : K . Al-Ghurabi
be bijective mapping the following statement
are equivalent :
a) F is homeomorphism .
b) F is continuous and open .
c) F is continuous and closed .
Proof :
(a
b ) → Assume (a) , let g be the inverse mapping of f so
&
→
, since g is continuous .
(G) is v-open &
(G) = f(G) is v-open , so f is open mapping .
← Assume (b) , i.e f is bijective by hypothesis & cont. open T.p
is cont.
Let G be T-open → f(G) is v-open →
(G) is v-open so
is
cont. → f is homeomorphism .
(c) → (a) . assume (c) T.p
is cont. let G be any T-open → X-G
is T-closed , since f is closed → f(X-G) =
→
(G) is v-open . Thus
(X-G) = y-
(G) is closed
is cont.
(a)→(c) , assume (a) , let H be T-closed → X-H is T-open and
is cont. →
(X-H) = y(H) is v-open
(H) is v-closed &
(H)= f(H) is v-closed , Hence f is cont.
III