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**Sample text**

I Figure 4,1 6. The eitherjor topology. This is another unusual topology in the plane. The neighborhoods are either disks that do not intersect the x-axis or the whole plane minus a finite number of points of the x-axis. 7. Compactification of the plane. Let £f be the plane plus an extra point called oo. Neighborhoods are the usual disks in the plane plus neighborhoods of oo consisting of oo together with the outside of the closed disks in the plane. This space is topologically equivalent to a sphere.

We will prove the following interesting corollary instead. Theorem A subset S of the plane is compact if and only if S is closed and bounded. We have already established that compact sets are closed and bounded. Now we must prove the converse. Therefore suppose that the set S is closed and bounded. Let {Pn} be a sequence in S. To prove that S is compact we must find a near point for {Pn} in S. Because S is bounded, there is a rectangle R enclosing S. The rectangle R is a cell, hence compact by the BolzanoWeierstrass theorem.

26 BASIC CONCEPTS It suffices to prove that just one particular path is connected, since both connectedness and the property of being a path are topological properties. Therefore consider the unit interval on the x-axis. Suppose that this interval is divided into two nonempty pieces A and B: A u B = [0, 1] and A n B = 0 . To prove connectedness, we must find a point in one of these sets near the other set. The proof uses a technique called the method of bisection. Assume that the endpoints of the interval come from both of the sets A and B.