Opšta topologija

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,^[1]^[2] geometrijsku topologiju^[3] i algebarsku topologiju.^[4]^[5]^[6] Drugi naziv za opštu topologiju je topologija skupa tačaka.

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

Neprekidne funkcije, intuitivno, prenese obližnje tačke do obližnjih tačaka
Kompaktni skupovi su oni koji mogu da budu pokriveni sa konačno mnogo skupova proizvoljno male veličine.
Povezani skupvi su skupovi koji se ne mogu podeliti u dva dela koja su daleko jedan od drugog.

Reči 'obližnji', 'proizvoljno mali', i 'daleko razdvojeni' se mogu učiniti preciznim koristeći koncept otvorenih skupova. Ako se promeni definicija 'otvorenog skupa', menja se ono što su neprekidne funkcije, kompaktni skupovi, i povezani skupovi. Svaki izbor definicije za 'otvoreni skup' se naziva topologija. Skup sa topologijom se naziva topološki prostor.

Metrički prostori su važna klasa topoloških prostora gde realna, nenegativna rastojanja, koja se takoše nazivaju metrici, mogu da budu definisana na parovima tačaka u skupu. Postojanje metrika pojednostavljuje mnoge dokaze, a mnogi najčešćih topoloških prostora su metrički prostori.

Istorija уреди

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

detaljno proučavanje podskupova realne linije (koja je nekada bila poznata kao topologija skupova tačaka; ova upotreba je sada zastarela)
uvođenje koncepta mnogostrukosti
proučavanje metričkih prostora, posebno normiranih linearnih prostora, u prvim danima funkcionalne analize.

Opšta topologija je svoj današnji oblik poprimila oko 1940. godine. Ona obuhvata, moglo bi se reći, gotovo sve unutar intuicije kontinuiteta, u tehnički adekvatnom obliku koji se može primeniti u bilo kojoj oblasti matematike.

Topologija na skupu уреди

Neka je X skup i neka je τ familija podskupova od X. Onda se τ naziva topologijom na X ako:^[7]^[8]

Prazan skup i X su elementi iz τ
Svaka unija elemenata iz τ je element iz τ
Svaki presek konačno mnogo elemenata iz τ je element iz τ

Ako je τ topologija na X, onda se par (X, τ) naziva topološkim prostorom. Notacija X_τ se može koristiti za označavanje skupa X na kome je primenljiva određena topologija τ.

Članovi τ se nazivaju otvorenim skupovima u X. Za podskup od X se kaže da je zatvoren, ako je njegov komplement u τ (i.e., njegov komplement je otvoren). Podskup od X može da bude otvoren, zatvoren, oba (zatvoreno-otvoren skup), ili ni jedno. Prazan skup i samo X su uvek otvoreni i zatvoreni.

Baze topologije уреди

Baza B za topološki prostor^[9]^[10] X sa topologijom T je kolekcija otvorenih skupova u T takvih da svaki otvoreni skup u T može da bude napisan kao unija elemenata od B.^[11]^[12] Kaže se da baza generiše topologiju T. Baze su korisne jer se mnoga svojstva topologija mogu redukovati do izjava o bazama koje generišu tu topologiju — i zato što se mnoge topologije najlakše definišu u pogledu baza koja ih generiše.

Reference уреди

^ Bott, R. and Tu, L.W., 1982. Differential forms in algebraic topology (Vol. 82, pp. xiv+-331). New York: Springer.
^ Milnor, J. and Weaver, D.W., 1997. Topology from the differentiable viewpoint. Princeton university press.
^ „What is geometric topology?”. math.meta.stackexchange.com. Приступљено 30. 5. 2018.
^ Fréchet, Maurice; Fan, Ky (2012), Invitation to Combinatorial Topology, Courier Dover Publications, стр. 101, ISBN 9780486147888 .
^ Henle, Michael (1994), A Combinatorial Introduction to Topology, Courier Dover Publications, стр. 221, ISBN 9780486679662 .
^ Spreer, Jonathan (2011), Blowups, slicings and permutation groups in combinatorial topology, Logos Verlag Berlin GmbH, стр. 23, ISBN 9783832529833 .
^ 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.
^ Schubert, Horst (1968), Topology, Macdonald Technical & Scientific, ISBN 0-356-02077-0
^ Sutherland, W. A. (1975). Introduction to metric and topological spaces. Oxford [England]: Clarendon Press. ISBN 0-19-853155-9. OCLC 1679102.
^ 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 уреди

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„Topological Shapes and their Significance”. arXiv:abs/1905.13481  Проверите вредност параметра |arxiv= (помоћ). 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.
Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966). ISBN 0-387-19374-X
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Spoljašnje veze уреди

The arXiv subject code is „math.GN”. arXiv:list/math.GN/recent  Проверите вредност параметра |arxiv= (помоћ). .

[1] Bott, R. and Tu, L.W., 1982. Differential forms in algebraic topology (Vol. 82, pp. xiv+-331). New York: Springer.

[milnor-2] Milnor, J. and Weaver, D.W., 1997. Topology from the differentiable viewpoint. Princeton university press.

[3] „What is geometric topology?”. math.meta.stackexchange.com. Приступљено 30. 5. 2018.

[4] Fréchet, Maurice; Fan, Ky (2012), Invitation to Combinatorial Topology, Courier Dover Publications, стр. 101, ISBN 9780486147888 .

[5] Henle, Michael (1994), A Combinatorial Introduction to Topology, Courier Dover Publications, стр. 221, ISBN 9780486679662 .

[6] Spreer, Jonathan (2011), Blowups, slicings and permutation groups in combinatorial topology, Logos Verlag Berlin GmbH, стр. 23, ISBN 9783832529833 .

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

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

[9] Schubert, Horst (1968), Topology, Macdonald Technical & Scientific, ISBN 0-356-02077-0

[10] Sutherland, W. A. (1975). Introduction to metric and topological spaces. Oxford [England]: Clarendon Press. ISBN 0-19-853155-9. OCLC 1679102.

[11] 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.”

[12] 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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