18. Continuous FunctionsDefn: f−1(V ) = {x | f(x) ∈ V }.Defn: f : X → Y is co ntinuous iff for every Vopen in Y , f−1(V ) is open in X.Lemma: f continuous if and only if for everybasis element B, f−1(B) is open in X.Lemma: f continuous if and only if for everysubbas is element S, f−1(S) is o pen in X.Thm 18.1: Let f : X → Y . Then the followingare equivalent:(1) f is continuous.(2) For every sub set A of X, f(A) ⊂ f(A).(3) For every closed set B of Y , f−1(B) is closedin X.(4) For each x ∈ X and each neighborhood V off(x), there is a neighborhood U of x such thatf(U) ⊂ V .15Defn: f : X → Y is a ho meomorphism iff f is abijection and both f and f−1is continuous.Defn: A property of a space X which is p reservedby homeomorphisms is called a topological prop-erty of X.Defn: f : X → Y is an imbedding of X in Y ifff : X → f (X) is a homeomorph ism.Thm 18.2(a.) (Constant function) The constant mapf : X → Y , f(x) = y0is continuous.(b.) (Inclusion) If A is a subs pace of X, then theinclusion map f : A → X, f(a) = a is continu-ous.(c.) (Composition) If f : X → Y and g : Y → Zare continuous, then g ◦f : X → Z is continuous.(d.) (Restricting the Domain) If f : X → Y iscontinuous and if A is a subsp ace of X, then therestricted function f|A: A → Y , f|A(a) = f (a)is continuous.16(e.) (Restricting or Expand ing the Codo ma in) Iff : X → Y is continuous and if Z is a sub spaceof Y containing the image set f(X) o r if Y is asubspa ce of Z, then g : X → Z is continuous.(f.) (Local formulation of continuity) Iff : X → Y and X = ∪Uα, Uαopen wheref|UαUα→ Y is continuou s, then f : X → Yis continuo us.Thm 1 8.3 (The pasting lemma): Let X = A ∪ Bwhere A, B are clos ed in X. Le t f : A → Y andg : B → Y be continuous. If f(x) = g(x) fo r allx ∈ A ∩ B, then h : X → Y ,h(x) =f(x) x ∈ Ag(x) x ∈ Bis continuo us.Thm 18.4: Let f : A → X × Y be given by theequations f(a) = (f1(a), f2(a)) wheref1: A → X, f2: A → Y . The n f is continu-ous if and only if f1and f2are continuous.17Defn: A group is a set, G, together with a func-tion ∗ : G × G → G, ∗(a, b) = a ∗ b such that(0) Closure: ∀a, b ∈ G, a ∗ b ∈ G.(1) Associativity: ∀a, b, c ∈ G,(a ∗ b) ∗ c = a ∗ (b ∗ c).(2) Identity: ∃ e ∈ G, such that ∀a ∈ G,e ∗ a = a ∗ e = a.(3) Inverses: ∀a ∈ G, ∃a−1∈ G such thata ∗ a−1= a−1∗ a = e.Defn: A group G is commutative or abelian if∀a, b ∈ G, a ∗ b = b ∗ a.Defn: A topological group is a set, G, such that(1) G is a gro up.(2) G is a topological space which is T1.(3) ∗ : G × G → G, ∗(a, b) = a ∗ band i : G → G, i(g) = g−1are both c ontinuous
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