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diff-manifolds

# We then have the definition def a function f x y is

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Unformatted text preview: ion: Def.: A function f : X Y is called continuous if f -1(G) is open in X for every open set G in Y . We can also define continuity at a point. Suppose f : X Y , x X, and y = f (x). We say that f is continuous at x if for every neighborhood V of y, there is a neighborhood U of x with f (U ) V . We then say that a function f is continuous if it is continuous at every x X. A homeomorphism from a topological space X to a topological space Y is a 1-1, onto, continuous function f : X Y whose inverse is also continous. 6 A topological space is called separable if there is a countable collection of open sets such that every open set in T can be written as a union of members of the countable collection. A topological space X is called Hausdorff if for every x, y X with x = y, there are neighborhoods U and V of x and y (respectively) with U V = . This is just the barest beginnings of Topology, but it should be enough to get us off the ground . . . 7 Exercises: Topological spaces 1. Show that the intersection of a finite numb...
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