Изворни код странице Opšta topologija

[[Image:Topologist's sine curve.svg|thumb|420px|[[Topologist's sine curve|Topologova sinusna kriva]] je koristan primer topologije skupa tačaka. Ona je povezana, ali nije putom povezan.]]

U [[mathematics|matematici]], '''opšta topologija''' je grana [[topology|topologije]] koja se bavi osnovnim definicijama i konstrukcijama [[Теорија скупова|teorije skupova]] koje se koriste u topologiji. Ono je osnova za većinu drugih grana topologije, uključujući [[differential topology|diferencijalnu topologiju]],<ref>[[Raoul Bott|Bott, R.]] and Tu, L.W., 1982. Differential forms in algebraic topology (Vol. 82, pp. xiv+-331). New York: Springer.</ref><ref name="milnor">Milnor, J. and Weaver, D.W., 1997. Topology from the differentiable viewpoint. Princeton university press.</ref> [[geometric topology|geometrijsku topologiju]]<ref>{{cite web |url=https://math.meta.stackexchange.com/questions/2840/what-is-geometric-topology |title=What is geometric topology? |website=math.meta.stackexchange.com |access-date=May 30, 2018}}</ref> i [[algebraic topology|algebarsku topologiju]].<ref>{{citation|title=Invitation to Combinatorial Topology| first1=Maurice|last1=Fréchet|author-link1=Maurice René Fréchet| first2=Ky|last2=Fan|author-link2=Ky Fan|publisher=Courier Dover Publications|year=2012|isbn=9780486147888|page=101|url=https://books.google.com/books?id=dfLSzs0vHNQC&pg=PA101}}.</ref><ref>{{citation|title=A Combinatorial Introduction to Topology|first=Michael|last=Henle|publisher=Courier Dover Publications|year=1994|isbn=9780486679662|page=221|url=https://books.google.com/books?id=S0RJYkBxowEC&pg=PA221}}.</ref><ref>{{citation|title=Blowups, slicings and permutation groups in combinatorial topology|first=Jonathan|last=Spreer|publisher=Logos Verlag Berlin GmbH|year=2011|isbn=9783832529833|page=23|url=https://books.google.com/books?id=vdt5c07k0M4C&pg=PA23}}.</ref> Drugi naziv za opštu topologiju je ''topologija skupa tačaka''.

Fundamentalni koncepti u opštoj topologiji su ''kontinuitet'', ''kompaktnost'', i ''povezanost'': 
* [[Continuous function|Neprekidne funkcije]], intuitivno, prenese obližnje tačke do obližnjih tačaka
* [[Kompaktan prostor|Kompaktni skupovi]] su oni koji mogu da budu pokriveni sa konačno mnogo skupova proizvoljno male veličine.
* [[Connected space|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 [[open set|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 ''[[topological space|topološki prostor]]''.

''[[Metric space|Metrički prostori]]'' su važna klasa topoloških prostora gde realna, nenegativna rastojanja, koja se takoše nazivaju ''[[metric (mathematics)|metrici]]'', mogu da budu definisana na parovima tačaka u skupu. Postojanje metrika pojednostavljuje mnoge dokaze, a mnogi najčešćih topoloških prostora su metrički prostori.

== Istorija ==

Opšta topologija je proizašla iz brojnih oblasti, najvažnije od kojih su:
* detaljno proučavanje podskupova [[real line|realne linije]] (koja je nekada bila poznata kao ''topologija skupova tačaka''; ova upotreba je sada zastarela)
* uvođenje koncepta [[mnogostrukost]]i
* proučavanje [[metric space|metričkih prostora]], posebno [[normed linear space|normiranih linearnih prostora]], u prvim danima [[Funkcionalna analiza|funkcionalne analize]].

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

== Topologija na skupu ==
{{Main-lat|Topološki prostor}}

Neka je ''X'' skup i neka je -{''τ''}- [[Family of sets|familija]] [[subset|podskupova]] od ''X''. Onda se -{''τ''}- naziva ''topologijom na X'' ako:<ref>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref><ref>Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</ref>

# [[Prazan skup]] i ''X'' su elementi iz ''τ''
# Svaka [[Унија (теорија скупова)|unija]] elemenata iz ''τ'' je element iz ''τ''
# Svaki [[intersection (set theory)|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<sub>τ</sub>'' se može koristiti za označavanje skupa ''X'' na kome je primenljiva određena topologija -{''τ''}-.

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

=== Baze topologije ===
{{Main-lat|Baza (topologija)}}

'''Baza''' -{''B''}- za [[topological space|topološki prostor]]<ref>{{citation|first=Horst|last=Schubert|title=Topology|year=1968|publisher=Macdonald Technical & Scientific|isbn=0-356-02077-0}}</ref><ref>{{Cite book|last=Sutherland|first=W. A.|url=|title=Introduction to metric and topological spaces|date=1975|publisher=Clarendon Press|isbn=0-19-853155-9|location=Oxford [England]|oclc=1679102}}</ref> ''X'' sa [[topological space|topologijom]] -{''T''}- je kolekcija [[open set|otvorenih skupova]] u -{''T''}- takvih da svaki otvoreni skup u ''T'' može da bude napisan kao unija elemenata od -{''B''}-.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |authorlink2=Howard Ensign Simmons, Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=https://archive.org/details/topologicalmetho00merr/page/16 |accessdate=27. 7. 2012 |pages=[https://archive.org/details/topologicalmetho00merr/page/16 16] |quote='''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''. |url-access=registration }}</ref><ref>{{cite book |last=Armstrong |first=M. A. |title=Basic Topology |year=1983 |publisher=Springer |isbn=0-387-90839-0  |url=https://www.springer.com/mathematics/geometry/book/978-0-387-90839-7 |accessdate=13. 6. 2013 |page=30 |quote=Suppose we have a topology on a set ''X'', and a collection <math>\beta</math> of open sets such that every open set is a union of members of <math>\beta</math>. Then <math>\beta</math> is called a ''base'' for the topology...}}</ref> 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 ==
{{reflist}}

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{{Литература крај}}

== Spoljašnje veze ==
{{Commons category-lat|General topology}}
* The [[arXiv]] subject code is {{cite journal |author= |date= |title=math.GN |url= |journal= |volume=|issue= |pages= |doi= |arxiv=list/math.GN/recent}}.

{{L|Области математике}}
{{L|Топологија}}

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