???????

??????? (Topology) ( ????????? τ?πο?, ???? , ??????? λ?γο?, ??????? ) ?????????? ??? ????? ???????. ?????????? ???? ???????????????? ??????????????? ????????? ???????? ?????? ???????? ???????? ???????????? ??? ???????????? ????? ??????? ??????????????. ???????????? Topology ???????, ????????, ????????? ????????? Topologie ??????? ????? ????????. ^[1]^[2]^[3]

??????? [ ???? ]

?????????? 1736 ??? ?????? (Euler), 1895 ??? ?????????? (Poincare), 1906 ??? ??????? (Frechet) , 1914 ??? ??????????? (Hausdorff) , 1922 ??? ??????????? (Kuratowski) ???????????? ??????? ??????? ????? ?????????? ????????????? ???????? ??????? ??????????. ^[4]^[5] ??????????? ??? ??????? ???????????????? ??????????? ???? ?????????? ????????????? ??????????? ??????? ????????. ????????????? ?????????? ?????? ????????????? ???????? ?????????????: ??????, ?????????????? ??????? (Set-theoretic Topology), ???????? ??????? (Algebraic Topology). ????????? ??????? ??????? ???? (Toplogical Space) ????? ?????????, ????????? ??????? ??? ?????????????? (Topological transformations) ????????? ????????????????? (Invariant characteristics) ????????? ??????????. ^[6]

?????????? ??????? (Intuitive Topology) [ ???? ]

?????????? ??????????? ?????? ????????? ????????? ??????? (equivalent) ????????? ????????? ??????????? ??? ?????????, ???????????, ???????? (?????????????) ??????? ???????? ?? ???? (Isometric) ???????????????? ??? ?????? ?????????? ???????? ???????????? ????????? ?????? ????? ??????????. ????????? ????????????? ????????? ???????????? ????????????? ????????? ???????????? ????????? ??????.

??? ??????? ??????????????? ???? ?????????. ?????????? ???? ?????????? ??? ?????? ?????? ??????? ??????????? ??? (??????, ?????????? ????????? ‘ ????? ’) ????? ????????????? ?????????? ??????????? ????????? ????????? ????? ????????????. ???????????????????? ??????? ?????????? ????? ?????.

??????? ??????? ?????????? ??????, ??????? ?????????? ?????????? ?????????? ???????? ???????? ??????????????? ??????? ????????? ?????? ????????????????. ???????? ?????????????????? ??????????? ???? ?? ???? ????????????, ?????????, ???????????? ???????????, ?????? ??????????? ??????????? ????? ????????? ??????? ????? ??????????? ??????????????????. ??? ?????????. ?????? ???? ?????? ??????????????? ??????? ???? ?????? ?????, ?????? , ???? ??????? ???????? ????????????? ???????? ???????? ???????????? ?????????? ????? ???????????. ?????????????? ??? ?????? ????????????? ????????? ????????, ???????? ????????????.

??????? ??????? [ ???? ]

???????? ???? S ??? ??????? ??????? ???????????? ???????? ???????? ??? ??????????? ?????????????? ???????? ?????????. ??????????????? ?????? ?????? ?????. ????? ??? ??????? ????????? ?????. S ??? ??????? ??????? ??????? X ??????, ??????? ??????? ??????? s ??????, “ s , ???? X ???? ??????? ?????????” ???? ?????????? ????? ?????????? ???????? ??????? ?????????????????, S ??? ??????? ??????? ???????????? ????? ?????????. ??, ???? S ??????? ????? ?????????????? ?????????? ?????????????????. ???? ???????????????? ?????? ???????????? ?????.

???? s ?? ????? X ?????? ??????? ?????????? ????? ???????? ?????????. ??????? S ??????? ???????? ??????????? ????????? , ?????????? , ????? ????????????. ??? ??? ?????????. ??? ???? ????????????. ??? ???????????? Trivial Topology ??????????.??????? ?????? ??????????? ??????.

???????????, ??? s ??? ??? ??????? ??????? X ???? ????????????? ?????, ???? ???????? ???????????. ??????? S ??????? ???????? ????????????????? ??????? ??????????? ????? ?????. ??????? ????????????.??? ???????????? Discrete Topology ??????.

?????????? ???????????? ?????? ???? ?? ????????????? ???????? ?????????????? ???? ?????? ????????????, ?????????????. ??? ?????????? ????????????? ??????? ??????.

??????? ????????????? ????????????????? ?????????? ???????????????, ????????????? ????? R ??? ??????? ??????? X ??????, ??????? ??????? s ?????? ????? ??????? ????????? ????????? ?????? ?????????????? ????????. ?????? R ??? ??? ??????????? ??????????????? ???????. ????????????? ??????????????? ????????? (??????, ???????? ) ?????????????? ??????????????. ?? ?????????????? ???? ?????? R ???? ?????? ???????????????? ???? ???????????? ????????????????; ????? ???? ???? ??????????????????????? ????????????!

?????? ????????? ????? ??????? ??????? [ ???? ]

???????? ???? S ? ??????????? ??????????? ????????????? ??? ??????? ??????????. ?????? ??????? ???? ??????? ????? ??? ?????? ??? ??????????. S ?????? ???????????? ??? ????????? ???????? ?????? ?????? ??????? ?????, ?????????? ( ??1, ??2, ??3 ????) ?????? ???????????????? ????? ???????? ?????? ???? ???? ???. (?? = ??????????)

( ?? 1 ): ??????????????, ?????????? S ??, ?????? ???????;

( ?? 2 ):??????? ?????? ??????? ??????????????????? ???????? ????????? , ?????? ??????????? ????????;

( ?? 3 ): ??? ????????? ??? ???????? ?????? ??????? ????????? ???????? ???????? ?????? , ?????? ??????????? ????????.

???????? ??????????????? ??????? ???????? ??????? ?????????? ?????? ?????? ?????. a ???? ??? ???????, ??????? X ???? ??????? ????????? ?????????????????? a ? ?????????? ??????? ?????? ?????? a ??????? X ??? ?????? ??????????? ???????????? ????????.

?????? R ??? ?????? ??????????????? ( a ? ε , a + ε ) ????? ?????? ?????????? ??????? a ? ?????????? ‘?????? ???????’ ???????.

???? ??????? ????????, ??????????????, ???????????? ?? ?????? ??????? ???? ?????? ???????. ?????? ????? ??????????? ????? ???????????????? ??????? ?????????. ???? ???????????, ??????? ??????? ???????? ??? ?????????? ???? ??????????????? ??????? ?????? ??????.

???????? ????? ??????????? [ ???? ]

?????? ??????? ????? ?????????????? ??????????? S ??????? ????? ???? ??????? ?????? ???????????????????. ???? S ??? p ???? ???????? ?????????? ?????????????????? ??? ?????????? ???? ????????? p ??? ???????????. ???????? ??????? p ?????? ???????? ????????. ??? p ??? ????? ?????????? ?????? A ?????????? ?????? ???????????????? ?????????????????: ( ? = ?????)

( ? 1 ) : S ε A ;,

( ? 2 ): A ???? ???? A ??? ????????? A ??? ??????? ???????????? A ??? ???????? ????????;

( ? 3 ): A , B ???????? A ??? ????????? A ∩ B ?? A ??? ???????? ????????.

???? ?????????, ???????? ????? ??????? ??????? ??????????????? S ??? ??????? ???????? ??????? p ??????, p ? ?????????? ????? ??????????????? ??????????? ( ?1 ), ( ?2 ), ( ?3 ) ????????????? ??? ?????? A ???????????? ??????? p ??? ???????? ????? ???????? ????????????? ??????? ???? ?????????????.

????????? ???????? ???????????? ????? ??????? ??????? ??????????? ?????, ?????????????? ???????? ???, ?????? ??????? ???, ??????? ????????? ??? ??????????? ?????????????? ???????? ???????? ??????? ???? ??????? ???????? ?????? ??????????. ???????? ????????? ????????? ????????? ????? ?????.

??????? ?????? [ ???? ]

???????????????? ???? ??????????? ?????????. S , T ???? ?????? ??????? ??????? ????????? ????????. S ????????? T ?????? ???????? ??????????????? (Transformations) ??????????. ???????? ???, ??????? ????????? ??????? ????????; ??????, ??????? ???????????? ??????????????. ???????? ??????? ????????? ???????? ?????????????? ????????????? ????????????????? (Continuous Trans-formations) ?????? ?????????????? ????????????? ??????. S ??, T ?? ????????????? ????????????, ?????? ??????? ??????????? ?????????, ?????? ????????????????? ???????? ????? ????????? (Continuous Functions). ?????? ???? ??????? ????????? ???????? ??????????????.

????????????????????, ?????? ??????? ??????? ??????????? (one-one maps), ??????? ?????? ??????????? (onto maps), ??????? ???????? ????????????? (Inverses) ????????? ?????????? ????????? ??? ??????????????????? (Topological Transformations) ????? ????? ??????.

???? ?????? ???????????????? ???????? ??? ???????????????? ?????????? ???? ??????? ??????? ????????? ????????? (equivalent) ???????????. ?????? ????????? ????????????? ???????????? ?????. ??????? ??????? ?????? ???? ???????. ?????????????? ????????????, ?????? ???????????? ????????? ??? ?????????? ?????????????????? ??? ??????? ???????? ????????????????????. ??????? ????????? ??? ??????? ????????? ???????? ?????????????? ?????????. ?????? ????????? ???????? ????????. ???????? ????????????? ??????? ???????????? ??????? ???????? ???????? ????????????? ???????? ????????????? ??????? ?????????????? ????????????? ????????? ?????????. ??? ???????????, ???? ?????????? ?????.

??????? ????????????????? ??????? ????? ???????? ??????????? . ???????????? Homeomorphism. ?????? ??????, ???????????? ?????????, ??????????? ?????????????? ???????? ???????????? ?????????????. ????? ?????? ??????????? ????????????. ???????? ?????? ??????????? ??????? ???? ??? ??????? ?????? ???????? ?????????????.

??????? ?????? ?????? ??? ???? ???? . ?????? ????? ??????? ????????? ?????? ??????????? ??????????????????. ??????? ??????????????? ????????? ??????? ????????????? ??????? ??????? ?????????. ?????? ????? ??????? ??????????? ?????????? ???????????? ???? ?????????? 19??? ???????????? ???????? ????? ???????? ??????????. ????????? ?????????????? ??????? ????? ???? . (Homology Theory). ??????? ???? ??????? ???? ???? (Homotopy Theory). ???????????? ??????? ???? ??????????? ???????????????? ????????? ??????????. ?????????? ??????????? ?????????????????? ??? ??????? ?????????? ??????? ??????? ??????: ??????-?????????? ?????????? (Euler-Poincare characteristic).

??????? ????????? ???????? [ ???? ]

????????? ???????????? ?????????? ??????? ???????? ????????? ??????????? ??????????? ???? ??????? ????? ?????????? ?????????. ????????? ???? ?????????? ???????? ????????? ??????, ??????, ???????, ???? ???????????? ?????????? (?????????? ??????????????? ??????????????? ???????), ????????? (??? ??????????????? ??????? ??? ???????????? ???? ?????????), ?????????????? ???????????, ?????????????, ????????????????? ??????????????, ?????????????? ??????, ??????? ?????? ???????????? ??????? ???????? ??????? ????????? ???????? ??????????? ?????????? ??????? ???????? ????????????? ???????????????. ??????, ?????? ??????? ???????? ???????? ???? ???? ???? ????? ????????????????, ?????? ?????? ??????????, ?????????? ??????????? ?????????? ???, ???????????? ????????????? ????????(?????????) ‘??????’?????? ????????????, ????????? ?????????????? ??????????????? ?????????? ??????? ??????????????. ????????? ???????????? ?????? ?????? ???????? ???????? ??????????? ????????? ??? ??? ??????? ?????????.

?????????? ??????????? [ ???? ]

???? ??????? [ ???? ]

History of Topology . (1999) Ed. I.M. James. Elsevier. Amsterdam.
Siefert, H and W. Threlfall. (1980) A Textbook of topology . Tr. By M.A. Goldman. Academic Press. New York,
V. Krishnamurthy. (1990). Culture, Excitement and Relevance of Mathematics . Wiley Eastern Ltd. New Delhi. ??????????? ?????????? ??? 81-224-0272-0

??????????? [ ???? ]

↑ "the definition of topology" .
↑ Richeson 2008 , ?. 63
↑ Aleksandrov 1969 , ?. 204
↑ Croom 1989 , ?. 7
↑ Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316?317
↑ Budney, Ryan (2011). "What is geometric topology?" . mathoverflow.net . ????????????? ???? 29 December 2013 .

??????????????? [ ???? ]

Hazewinkel, Michiel, ed. (2001), "Topology, general" , Encyclopedia of Mathematics , Springer, ??????????? ?????????? ??? 978-1556080104
Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.
??????? ?????? ????? ???????????
The Topological Zoo ????????????? 2012-02-04 at the ??????? ????????? at The Geometry Center .
Topology Atlas
Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.
Topology Glossary ????????????? 2009-07-13 at the ??????? ?????????
Moscow 1935: Topology moving towards America , a historical essay by Hassler Whitney .

[1] "the definition of topology" .

[2] Richeson 2008 , ?. 63

[3] Aleksandrov 1969 , ?. 204

[Croom7-4] Croom 1989 , ?. 7

[5] Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316?317

[6] Budney, Ryan (2011). "What is geometric topology?" . mathoverflow.net . ????????????? ???? 29 December 2013 .

[1]

[2]

[3]

[4]

[5]

[6]

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