Countably in a sentence

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers.

Every countably compact space is pseudocompact and weakly countably compact.

Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.

A complete binary tree * In the infinite complete binary tree, every node has two children (and so the set of levels is countably infinite ).

A distribution that has zero probability at countably many points on the real line and has a Gaussian shape everywhere else on the real line is likewise mesokurtic.

As noted above, this terminology is not universal.: Some authors use countable to mean what is here called countably infinite, and do not include finite sets.

A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets.

Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space.

Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points).

For example, suppose there were a largest ship, directly containing a finite (or countably infinite) number of both ships and people, and each of these ships in turn contained both ships and people, and so forth.

For the number of potential values to be countably infinite, even though their probabilities sum to 1, the probabilities have to decline to zero fast enough.

However, m is usually not countably additive, and hence does not define a measure in the usual sense.

In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite.

It must assign 0 to the empty set and be ( countably ) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets.

Kaplansky Commutative rings, p. 10, Ex 10. * In a commutative ring, an ideal maximal with respect to being not countably generated is prime.

Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).

Non-finitely generated abelian groups Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.

Observe that N in that theorem can be replaced with any countably infinite set.

Properties Several further properties can be derived from the definition of a countably additive measure.

Since X is not measurable for any rotation-invariant countably additive finite measure on S, finding an algorithm to select a point in each orbit requires the axiom of choice.