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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. I 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 : II 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