The extreme value theorem follows from the fact that this property is preserved by continuous functions.
Three different types of this property are equivalent on a Banach space according to the Eberlein-ÂŠmulian theorem. A method of imparting this property to a space by adjoining a single point to it is called the Alexandroff extension. Specific varieties of this property include the "limit point" variety and the "sequential" variety. A family of functions will not have this property without being (*) equicontinuous. The extreme value theorem follows from the fact that this property is preserved by continuous functions. This property, which is the subject of the Arzela-Ascoli theorem, is preserved under products according to Tychonoff's theorem. A set on which every open cover has a finite subcover possesses this property. For 10 points, name this property possessed by a set which is closed and bounded, by the Heine-Borel theorem.
Three different types of this property are equivalent on a Banach space according to the Eberlein-ÂŠmulian theorem.
compactness [accept weak compactness before "Alexandroff"]