In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base. More explicitly, this means that a topological space is second countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.
Most "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rn) with its usual topology is second-countable. Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.
Read more about Second-countable Space: Properties, Examples
Famous quotes containing the word space:
“The woman’s world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)