ONT Re: Topology
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Note 26
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With great reluctance I am going to skip Chapter 2 on "Convergence"
and proceed directly to Chapter 3 on "Product and Quotient Spaces".
I am doing this by way of more quickly picking up the all-important
ideas of a "continuous function" and a "homeomorphism", the latter
also known as a "topological transformation".
| 3. Product and Quotient Spaces
|
| It is the purpose of this chapter to investigate two methods of constructing
| new topological spaces from old. One of these involves assigning a standard
| sort of topology to the cartesian product of spaces, thus building a new space
| from those originally given. For example, the Euclidean plane is the product
| space of the real numbers (with the usual topology) with itself, and Euclidean
| n-space is the product of the real numbers n times. In chapter 4 arbitrary
| cartesian products of the real numbers will serve as standard spaces with
| which one may compare other topological spaces.
|
| The second method of constructing a new space from a given one depends on
| dividing the given space X into equivalence classes, each of which is a point
| of the newly constructed space. Roughly speaking, we "identify" the points of
| certain subsets of X, so obtaining a new set of points, which is then assigned
| the "quotient" topology. For example, the equivalence classes of real numbers
| modulo the integers are assigned a topology so that the resulting space is
| a "copy" of the unit circle in the plane.
|
| Both of these methods of constructing spaces are motivated by making certain
| functions continuous. We therefore begin by defining continuity and proving
| a few simple propositions about it.
|
| JLK, Gen Top, page 84.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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Topology
01. http://suo.ieee.org/ontology/msg03863.html
02. http://suo.ieee.org/ontology/msg03867.html
03. http://suo.ieee.org/ontology/msg03868.html
04. http://suo.ieee.org/ontology/msg03869.html
05. http://suo.ieee.org/ontology/msg03870.html
06. http://suo.ieee.org/ontology/msg03871.html
07. http://suo.ieee.org/ontology/msg03872.html
08. http://suo.ieee.org/ontology/msg03874.html
09. http://suo.ieee.org/ontology/msg03880.html
10. http://suo.ieee.org/ontology/msg03882.html
11. http://suo.ieee.org/ontology/msg03883.html
12. http://suo.ieee.org/ontology/msg03888.html
13. http://suo.ieee.org/ontology/msg03889.html
14. http://suo.ieee.org/ontology/msg03892.html
15. http://suo.ieee.org/ontology/msg03893.html
16. http://suo.ieee.org/ontology/msg03894.html
17. http://suo.ieee.org/ontology/msg03899.html
18. http://suo.ieee.org/ontology/msg03900.html
19. http://suo.ieee.org/ontology/msg03903.html
20. http://suo.ieee.org/ontology/msg03908.html
21. http://suo.ieee.org/ontology/msg03914.html
22. http://suo.ieee.org/ontology/msg03916.html
23. http://suo.ieee.org/ontology/msg03917.html
24. http://suo.ieee.org/ontology/msg03918.html
25. http://suo.ieee.org/ontology/msg03920.html
26. http://suo.ieee.org/ontology/msg03921.html
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