Opšta topologija — разлика између измена

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U matematici, opšta topologija je grana topologije koja se bavi osnovnim definicijama i konstrukcijama teorije skupova koje se koriste u topologiji. Ono je osnova za većinu drugih grana topologije, uključujući diferencijalnu topologiju, geometrijsku topologiju i algebarsku topologiju. Drugi naziv za opštu topologiju je topologija skupa tačaka.

Fundamentalni koncepti u opštoj topologiji su kontinuitet, kompaktnost, i povezanost:

Continuous functions, intuitively, take nearby points to nearby points.
Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
Connected sets are sets that cannot be divided into two pieces that are far apart.

The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

Metrički prostori su važna klasa topoloških prostora gde realna, nenegativna rastojanja, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

Istorija

Opšta topologija je proizašla iz brojnih oblasti, najvažnije od kojih su:

the detailed study of subsets of the real line (once known as the topology of point sets; this usage is now obsolete)
the introduction of the manifold concept
the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.

General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

Topologija na skupu

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:^[1]^[2]

Both the empty set and X are elements of τ
Any union of elements of τ is an element of τ
Any intersection of finitely many elements of τ is an element of τ

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X_τ may be used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open.

Baze topologije

Baza B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.^[3]^[4] We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.

Reference

^ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
^ Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
^ Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. стр. 16. ISBN 0-471-83817-9. Приступљено 27. 7. 2012. „Definition. A collection B of subsets of a topological space (X,T) is called a basis for T if every open set can be expressed as a union of members of B.”
^ Armstrong, M. A. (1983). Basic Topology. Springer. стр. 30. ISBN 0-387-90839-0. Приступљено 13. 6. 2013. „Suppose we have a topology on a set X, and a collection $\beta$ of open sets such that every open set is a union of members of $\beta$ . Then $\beta$ is called a base for the topology...”

Literatura

John L. Kelley (1955) General Topology, link from Internet Archive, originally published by David Van Nostrand Company.
George F. Simmons, Introduction to Topology and Modern Analysis, ISBN 1-575-24238-9.
Paul L. Shick, Topology: Point-Set and Geometric, ISBN 0-470-09605-5.
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 изд.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446
O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, Elementary Topology: Textbook in Problems, ISBN 978-0-8218-4506-6.
Topological Shapes and their Significance by K.A.Rousan arvXiv id- 1905.13481
Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate Texts in Mathematics. Springer. ISBN 0-387-90839-0.
Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997). ISBN 0-387-97926-3.
Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966). ISBN 0-387-19374-X
Brown, Ronald, Topology and Groupoids, Booksurge (2006) ISBN 1-4196-2722-8 (3rd edition of differently titled books)
Čech, Eduard; Point Sets, Academic Press (1969).
Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997). ISBN 0-387-94327-7.
Gallier, Jean; Xu, Dianna (2013). A Guide to the Classification Theorem for Compact Surfaces. Springer.
Gauss, Carl Friedrich; General investigations of curved surfaces, 1827.
Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.
Schubert, Horst (1968), Topology, Macdonald Technical & Scientific, ISBN 0-356-02077-0
Vaidyanathaswamy, R. (1960). Set Topology. Chelsea Publishing Co. ISBN 0486404560.
Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
Arkhangel’skij, A.V.; Ponomarev, V.I. (1984). Fundamentals of general topology: problems and exercises. Mathematics and Its Applications. 13. Translated from the Russian by V. K. Jain. Dordrecht: D. Reidel Publishing. Zbl 0568.54001.
Engelking, Ryszard (1977). General Topology. Monografie Matematyczne. 60. Warsaw: PWN. Zbl 0373.54002.

Spoljašnje veze

The arXiv subject code is math.GN.

[1] Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

[2] Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.

[3] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons. стр. 16. ISBN 0-471-83817-9. Приступљено 27. 7. 2012. „Definition. A collection B of subsets of a topological space (X,T) is called a basis for T if every open set can be expressed as a union of members of B.”

[4] Armstrong, M. A. (1983). Basic Topology. Springer. стр. 30. ISBN 0-387-90839-0. Приступљено 13. 6. 2013. „Suppose we have a topology on a set X, and a collection $\beta$ of open sets such that every open set is a union of members of $\beta$ . Then $\beta$ is called a base for the topology...”

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