Download PDF by Michael Henle: A Combinatorial Introduction to Topology (Dover Books on

By Michael Henle

ISBN-10: 0486679667

ISBN-13: 9780486679662

Very good textual content for upper-level undergraduate and graduate scholars exhibits how geometric and algebraic principles met and grew jointly into a major department of arithmetic. Lucid assurance of vector fields, surfaces, homology of complexes, even more. a few wisdom of differential equations and multivariate calculus required. Many difficulties and workouts (some ideas) built-in into the textual content. 1979 version. Bibliography.

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Additional info for A Combinatorial Introduction to Topology (Dover Books on Mathematics)

Example text

It may happen that some point P is brought back to its original position. 1 Continuous transformation of a cell into itself. moved, f(P) = P, and P is called a fixed point of / If it happens that every continuous transformation, no matter how violent, of D into D has a fixed point, then we say that D has the fixed point property. The fixed point property is a topological property. To prove this, let u: D -• R be a topological transformation from a set D with the fixed point property to a set R.

Here is an unusual topology on the plane. 1). ^ ! 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.

Since P is near the sequence {Pn}, P is also near or in the set S. Then because S is closed, it follows that P must be in S. This completes the proof. Exercises 1. Complete the proof that compact sets are bounded by showing that the sequence 0* constructed there has no near points. 2. Let the point P be near the set S. Show that S contains a sequence whose only near point is P. Use this result to prove the first theorem of this section and to prove that compact sets are closed. 3. , compact)? } (h) the whole plane (i) the empty set 4.

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A Combinatorial Introduction to Topology (Dover Books on Mathematics) by Michael Henle

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