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Basic Definitions 3 Exercise 4. Prove that the spaces W1 and Sn \ {xo} a r e homeomorphic. Problem 2. Prove that 5 n + m ~ 1 \ S12'1 « Rn x 5 m ~ 1 . (We assume that the sphere S'n_1 is standardly embedded in Sn+m~1.) A topological space X is said to be discrete if all of its subsets are open (or, equivalently, if all of its subsets are closed). The topology of a discrete topological space is called the discrete topology. If X is a discrete topological space and Y is an arbitrary topological space, then any map / : X -+ Y is continuous. A topological space X is called connected if it contains no proper subsets which are both open and closed. In other words, if a set A C X is both open and closed, then either A = 0 or A = X. Exercise 5. Prove that the space W1 is connected. Exercise 6. Prove that if X is a connected topological space and Y is a discrete topological space, then any continuous map / : X — Y is constant, i.e., f(X) consists of one point. Problem 3. Prove that if A and B are connected subsets of a topological space X that are not disjoint (i.e., have a nonempty intersection), then AUB is connected. Problem 3 shows that the following relation on the set of points of a topological space X is an equivalence: two points are considered equivalent if there exists a connected set containing both of them. The equivalence class of a point x G X is the maximal connected set containing x. It is called a connected component. A metric space is a set X such that, for any two points x,y G X, a number d(#, y) 0 is defined and satisfies the following conditions: (1) d{x,y) = d(y,x)\ (2) d(x,y) + d(y,z) d(x,z) (the triangle inequality) (3) d(x, y) — 0 if and only if x — y. The number d(x,y) is called the distance between the points x and y. In any metric space X, the open balls D%£ — {x G X : d(x, a) e} form a base for some topology. This topology is said to be induced by the metric d. If X is a topological space and its topology is induced by some metric, then X is metrizable. A topological space is said to be compact if any cover of this space by open sets has a finite subcover. Exercise 7. Prove that the sphere Sn is compact and the space IRn is noncompact.