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 11, 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...
View
Full Document
 Fall '09
 Carter
 Differential Equations, Logic, Topology, Equations, Metric space, Topological space, Differentiable Manifolds, topological spaces

Click to edit the document details