Resolvable space
Encyclopedia
In topology
, a topological space
is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because the rational
s and irrational
s are disjoint dense subsets. A topological space that is not resolvable is termed irresolvable.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because the rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s and irrational
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s are disjoint dense subsets. A topological space that is not resolvable is termed irresolvable.
Properties
- The productProduct topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
of two resolvable spaces is resolvable - Every locally compact topological space without isolated pointIsolated pointIn topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...
s is resolvable - Every submaximal space is irresolvable