Kompaktan prostor

U matematici, i specifičnije opštoj topologiji, kompaktnost je svojstvo koje generalizuje pojam podskupa Euklidovog prostora koji je zatvoren (da sadrži sve svoje granične tačke) i ograničen (onaj kod koga sve njegove tačke leže na datom fiksnom rastojanju jedna od druge). Primeri su zatvoreni interval, četvorougao, ili konačni set tačaka. Ovaj je pojam definisan za opštije topološke prostore, nego što je Euklidov prostor na razne načine.

Interval A = (−∞, −2] nije kompaktan zato što nije ograničen. Interval C = (2, 4) nije kompktan zato što nije zatvoren. Interval B = [0, 1] je kompaktan zato što je zatvoren i ograničen.

Jedna takva generalizacija je da je topološki prostor sekvencijalno kompaktan ako svaki infinitivni niz tačaka uzet kao uzorak prostora ima beskonačni podniz koji konvergira u istu tačku prostora. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1] some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (others accumulate to 1). The same set of points would not accumulate to any point of the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any real number.

Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th-century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem or the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

Literatura

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Spoljašnje veze

• Countably compact at PlanetMath.org.
• Sundström, Manya Raman (2010). „A pedagogical history of compactness”. arXiv:1006.4131v1  [math.HO].