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Then, for nonempty , the properties numerable (which here shall mean that injects into ), countable (has as its range), subcountable (a subset of surjects into ) and also not -productive (a countability property essentially defined in terms of subsets of ) are all equivalent and express that a set is finite or countably infinite.
IAS 16 applies to property, plant and equipment (PPE). The standard itself defines PPE as "tangible items that are held for use in the production or supply of goods or services, for rental to others, or for administrative purposes; and are expected to be used during more than one [accounting] period."
Here, every unbounded subset of is then in bijection with itself, and every subcountable set (a property in terms of surjections) is then already countable, i.e. in the surjective image of . In this context the possibilities are then exhausted, making " ≤ {\displaystyle \leq } " a non-strict partial order , or even a total order when assuming ...
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable open set. The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets.
The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. [67] In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. [68]
The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters.
Any countable product of a second-countable space is second-countable, although uncountable products need not be. The topology of a second-countable T 1 space has cardinality less than or equal to c (the cardinality of the continuum). Any base for a second-countable space has a countable subfamily which is still a base.