2 Basic Definitions and only if, for any (7, V G rf and any x £ U P\V, there exists a W G r' such that x G l f C t / n F . A topological space X is called second countable if it has a countable base. For example, the open balls JD™£, where the numbers e and all coor- dinates of the points a are rational, form a countable base of the space Rn. Problem 1. Let X be a second countable topological space. Prove that any cover of X by open sets Ua has a countable subcover. If X is a topological space and Y is a subset of X, then Y can be endowed with the induced topology, which consists of the intersections of Y with open subsets of X. This turns the sphere Sn = {x G R n + 1 : \\x\\ = 1} into a topological space. A map of topological spaces is said to be continuous if the preimage of any open set is open, or, equivalently, the preimage of any closed set is closed. In proving the continuity of a map / , the following continuity criterion is often convenient: A map f: X Y is continuous if and only if for any point x G X and any neighborhood U of f(x), there exists a neighborhood V(x) of x such that its image is entirely contained in U. Indeed, if this condition holds, then the preimage f~l(U) of any open set U can be repre- sented as the union Lbef-W) V(x) °f °P e n se ^s a n d 1S therefore open. The converse assertion is obvious: for V(x) we can take f~l(U). Exercise 3. Prove that a map / : Rn Rm is continuous if and only if, for any x G Rn and any e 0, there exists a.5 0 such that \\f(x) f(a)\\ e whenever ||x a\\ 6. The following gluing theorem for continuous maps is used fairly often in topology. Theorem 0.1. Suppose that X = X\ U U Xn and the sets X\, ..., Xn are closed. A map f: X -— Y is continuous if and only if the restrictions fi = f\Xi are continuous. Proof. Clearly, if a map / is continuous, then all of the maps fi are con- tinuous as well. Suppose that the maps fi are continuous and C C Y is an arbitrary closed set. Then, for each i, the set C{ = f~l(C) = f~1(C)P\Xi is closed in Xi, i.e., there exists a closed subset C[ of X such that C{ = C^flJQ. Both sets C[ and Xi are closed in X therefore, C{ is closed in X also. Hence the set f~\C) = Cx U U Cn is closed in X. A map / : X » Y is called a homeomorphism if it is one-to-one and both of the maps / and f~l are continuous. In this case, we say that the topological spaces X and Y are homeomorphic and write X « Y.
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