Reference for COCOUNTABILITY. Search for COCOUNTABILITY

AI searches containing COCOUNTABILITY

COCOUNTABILITY

Cocountability

Property of mathematical sets

In mathematics, a cocountable subset of a set X {\displaystyle X} is a subset Y {\displaystyle Y} whose complement in X {\displaystyle X} is a countable

Cocountability

Cocountable topology

Topology made of cocountable subsets

{\displaystyle X} whose complements are countable, a property known as cocountability. The only closed sets in this topology are X {\displaystyle X} itself

Cocountable topology

Cocountable_topology

General topology

Branch of topology

the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or

General topology

General_topology

List of examples in general topology

field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete

List of examples in general topology

List_of_examples_in_general_topology

Topological space

Mathematical space with a notion of closeness

the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or

Topological space

Topological_space

Compact space

Type of mathematical space

carrying the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact. Like the

Compact space

Compact_space

Cofiniteness

Subset with finite complement

complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to

Cofiniteness

Hausdorff space

Type of topological space

Hausdorff is the cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically

Hausdorff space

Hausdorff_space

Urysohn and completely Hausdorff spaces

Form of topological spaces

The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology

Urysohn and completely Hausdorff spaces

Urysohn_and_completely_Hausdorff_spaces

Sequential space

Topological space characterized by sequences

not Fréchet–Urysohn. The simplest space that is not sequential is the cocountable topology on an uncountable set. Every convergent sequence in such a space

Sequential space

Sequential_space

Compactly generated space

Property of topological spaces

anticompact T1 non-discrete space is not CG-1. Examples include: The cocountable topology on an uncountable space. The one-point Lindelöfication of an

Compactly generated space

Compactly_generated_space

Von Neumann algebra

*-algebra of bounded operators on a Hilbert space

for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set. A fundamental approximation theorem can

Von Neumann algebra

Von_Neumann_algebra

Filter on a set

Family of subsets representing "large" sets

{\displaystyle X} is a set, the cocountable subsets of X {\displaystyle X} (those whose complement is countable) form a filter, the cocountable filter which is finer

Filter on a set

Filter_on_a_set

List of topologies

List of concrete topologies and topological spaces

its complement are open. Cocountable topology Given a topological space ( X , τ ) , {\displaystyle (X,\tau ),} the cocountable extension topology on X

List of topologies

List_of_topologies

Luzin N property

Measure theory concept

Luzin N property. This extends to functions that are differentiable on a cocountable set, as the image of a countable set is countable and thus a null set

Luzin N property

Luzin_N_property

Separation axiom

Axioms in topology defining notions of "separation"

space is a KC space, but the reverse is not necessarily true; for the cocountable topology on uncountably many points, the compact sets are all finite

Separation axiom

Separation_axiom

Kuratowski closure axioms

Axioms for defining a topology

X} ; if λ = ℵ 1 {\displaystyle \lambda =\aleph _{1}} , it induces the cocountable topology. Since any Kuratowski closure is isotonic, and so is obviously

Kuratowski closure axioms

Kuratowski_closure_axioms

Generic property

Property holding for typical examples

one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes

Generic property

Generic_property

List of general topology topics

Discrete space Locally constant function Trivial topology Cofinite topology Cocountable topology Finer topology Product topology Restricted product Quotient

List of general topology topics

List_of_general_topology_topics

Limit point compact

Type of topological space in mathematics

limit point compact. An example is given by an uncountable set with the cocountable topology. Every normal pseudocompact space is limit point compact. Proof:

Limit point compact

Limit_point_compact