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2 Continuous maps. Definition 2.1 Let (X, U) and (Y, V) a function f : X → Y is called continuous if the inverse image of any open set in Y is an open set in X. As a formula the definition is. (1) f is continous ⇔ ∀(V ∈ V) f −1 (V ) ∈ U Theorem 2.2 Let (X, U) and (Y, V) be topological spaces, and f : X → Y a function. The following are equivalent. (i) f is continous ∀(x ∈ X) ∀(V ∈ N (f (x))) ∃(U ∈ N (x)) f (U ) ⊆ V (ii) ∀(A ⊆ X)f (A) ⊆ f (A) (iii) 3 Induced topologies. Definition 3.1a (The direct image topology) If f : X → Y is a function and U is a topology on X, the topology on Y induced by f is the topology whose open sets are exactly the sets V ⊆ Y , with the property that their inverse image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct image topology. V ∈ f∗ (U) ⇔ f −1 (V ) ∈ U (1a) Definition 3.1b (The inverse image or pull-back topology) If f : X → Y is a function and V is a topology on Y , the topology on X induced by f is the topology whose open sets are exactly the sets of the form f −1 (V ) for V ∈ V. This topology is denoted f ∗ (U). The formula below defines the pull-back topology. U ∈ f ∗ (V) ⇔ ∃(V ∈ V) U = f −1 (V ) (1b) The following are facts about induced topologies. The proofs are left to the reader. Theorem 3.2 Let (X, U) and (Y, V) be topological spaces, and f : X → Y a function. The following are equivalent. (i) f : (X, U) → (Y, V) (ii) f∗ (U) ⊇ V (iii) is continous U ⊇ f ∗ (V) Corollary 3.3a If f : X → Y is a function and U is a topology on X, then the topology on Y induced by f is the finest topology rendering f continuous. In other words. (3a) f∗ (U) = sup{W | f : (X, U) → (Y, W) is continous} Corollary 3.3b If f : X → Y is a function and V is a topology on Y , then the topology on X induced by f is the coarsest topology rendering f continuous. In other words. (3b) f ∗ (V) = inf{W | f : (X, W) → (Y, V) 1 is continous}