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By Allan J. Sieradski

The remedy of the topic of this article isn't encyclopedic, nor was once it designed to be appropriate as a reference guide for specialists. relatively, it introduces the themes slowly of their historical demeanour, in order that scholars will not be crushed via the last word achievements of numerous generations of mathematicians. cautious readers will see how topologists have steadily subtle and prolonged the paintings in their predecessors and the way such a lot sturdy rules achieve past what their originators expected. To motivate the advance of topological instinct, the textual content is abundantly illustrated. Examples, too a variety of to be thoroughly lined in semesters of lectures, make this article appropriate for self sustaining examine and make allowance teachers the liberty to choose what they'll emphasize. the 1st 8 chapters are appropriate for a one-semester path normally topology. the full textual content is acceptable for a year-long undergraduate or graduate point curse, and offers a powerful starting place for a next algebraic topology path dedicated to the better homotopy teams, homology, and cohomology.

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Therefore rr - S is not connected. 2), S must be the boundary of each component of A - S. Thus each component of rr - S must meet C - (a + b) and hence must contain either C1 or C2. Therefore there are just two such components. 3) JORDAN CURVE THEOREM. Every simple closed curve J divides the plane rr into just two regions and is the boundary of each of these regions. Proof. There exists a straight line segment ab meeting J in exactly the two points a and b. Let axlb and axzb be the two arcs of J from a to b, and denote by Sl and S2, respectively, the semi-polygons ab + ax1b and ab + axtb.

31) Every elementary region i8 uniformly locally connected. DEFINITION. If A and B are closed 2-cells, or bounded elementary regions, then simple subdivisions S. and Sb of A and B respectively are said to be isomorphic or Similar provided there exists a similarity correspondence between them. By a similarity correspondence is meant a 1-1 relationship between S. and Sb, say h(SQ) = h(Sb), which maps the graph G. consisting of the union of the boundaries of the 2-cells of S. topologically onto the graph Gb made up of the union of the 2-cells of Sb and, in addition, establishes a 1-1 relation between the 2-cells of S.

Then the sequence (x*) is a fundamental sequence. For given e > 0, an N exists so that for in, n > N, p(xm, xA) = lim p(xk, x7) < e/3. k-. co Thus if N is chosen also so that 1/N < E/3, then for m, n > N we have p(xk, xk) < E/3 for a definite k sufficiently large. Whence p( m, S p(xm, xk) + p(xk , xk) + p(xk, 4) < E/3 + e/3 + e/3 = E. Accordingly p = (xn) is a point of X. ,,, p(xn, xn) < e, we have xk --, p so that X is complete. 2) THEOREM. Any metric space X can be isometrically imbedded in a complete space % (called the complete enclosure of X) in which X is dense.

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