Countable

A countable product of second countable spaces is second countable, but an uncountable product of second countable spaces need not even be first countable.

Separability versus second countability Any second-countable space is separable: if is a countable base, choosing any from the non-empty gives a countable dense subset.

So we are talking about a countable union of countable sets, which is countable by the previous theorem.

Theorem: (Assuming the axiom of countable choice ) The union of countably many countable sets is countable.

The above examples show that ‘some’ works with both countable and uncountable nouns – unlike ‘a few/few’ that goes with only countable (a few passengers).

A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets.

A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω 1 ( omega-one ), that is, if and only if the set is countable or has the smallest uncountable order type.

Definition Countable additivity of a measure μ : The measure of a countable disjoint union is the same as the sum of all measures of each subset.

Do these have fewer elements than N? Theorem: Every subset of a countable set is countable.

Every second-countable space is first-countable, separable, and Lindelöf.

If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous.

Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (and hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric.

Many nouns have both countable and uncountable uses; for example, beer is countable in "give me three beers", but uncountable in "he likes beer".

Proposition: Any subset of a countable set is countable.

Proposition: If A and B are countable sets then A ∪ B is countable.

Proposition: The Cartesian product of two countable sets A and B is countable.

Proposition: The integers Z are countable and the rational numbers Q are countable.

Since each step removes a finite number of intervals and the number of steps is countable, the set of endpoints is countable while the whole Cantor set is uncountable.

Surfaces that are not even second countable There exist (necessarily non-compact) topological surfaces having no countable base for their topology.

The last fact follows from f(X) being compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable.