By Theodor Bröcker, Klaus Jänich
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Extra info for Einfuehrung in die Differentialtopologie
Note also that an arbitrary subset U ⊆ M is in S iff U = i∈I Ui with Ui = Mi for all i ∈ I except for one i 0 ∈ I and Ui0 ∈ Mi0 . Thus for a finite intersection U of such sets we still have Ui = Mi for all but finitely many i ∈ I . Hence U is in B iff U is the product of Ui ’s with all Ui equal to Mi except for finitely many Ui1 ∈ Mi1 , . . , Uin ∈ Min . In the case of a finite Cartesian product M = M1 × · · · × Mn we note that the above basis of the product is simply given by B = O1 × · · · × On O1 ∈ M1 , .
Then there is an open O ⊆ U with f ( p) ∈ O and hence p ∈ f −1 (O) ∈ U( p), since f −1 (O) is open by continuity. But then f −1 (O) ⊆ f −1 (U ) shows f −1 (U ) ∈ U( p) giving (ii) =⇒ (i) Finally, the compatibility of and with preimages shows the equivalence of (ii) and (iv). In particular, the fourth part is often very convenient for checking continuity as we can use a rather small and easy subbasis instead of the typically huge and complicated topology. 3 Let f : (M, M) −→ (N , N ) and g : (N , N ) −→ (K , K) be maps between topological spaces.
Proof Let q ∈ M be a point with f (q) = g(q). Then the Hausdorff property implies that we find open subsets O1 , O2 ⊆ N with f (q) ∈ O1 and g(q) ∈ O2 but O1 ∩O2 = ∅. By continuity, f −1 (O1 ) and g −1 (O2 ) are open and q ∈ f −1 (O1 ) ∩ g −1 (O2 ). If q is another point in this intersection then f (q ) ∈ O1 and g(q ) ∈ O2 yielding f (q ) = g(q ) as O1 ∩ O2 = ∅. This shows the first part by taking complements. The second part is now easy as the set U ⊆ M is in the closed coincidence set and thus also U cl is in the closed coincidence set.
Einfuehrung in die Differentialtopologie by Theodor Bröcker, Klaus Jänich