Countably compact spaces

Definition

In topological theory, a branch of mathematics, a countably compact space is a type of topological space with certain compact-like properties. Specifically, a topological space is said to be countably compact if every countable open cover has a finite subcover. This is a weaker condition than compactness, which requires that every open cover, regardless of its cardinality, has a finite subcover.

Properties

Countably compact spaces have several important properties that distinguish them from other types of topological spaces.

Limit Point Compactness

A topological space is countably compact if and only if it is limit point compact. This means that every infinite subset of the space has a limit point in the space. This property is a direct consequence of the definition of countably compactness.

Sequential Compactness

A topological space is countably compact if and only if it is sequential compact. This means that every sequence in the space has a convergent subsequence. This property is not equivalent to countably compactness in general, but it is equivalent under certain additional assumptions, such as first-countability.

Pseudocompactness

A topological space is countably compact if and only if it is pseudocompact. This means that every continuous real-valued function on the space is bounded. This property is not equivalent to countably compactness in general, but it is equivalent under certain additional assumptions, such as Tychonoff's theorem.

Examples

Several types of topological spaces are countably compact.

Finite Spaces

Every finite topological space is countably compact. This is because every open cover of a finite space is countable, and therefore has a finite subcover.

Compact Spaces

Every compact space is countably compact. This is because compactness is a stronger condition than countably compactness.

The Cantor Set

The Cantor set, a classic example of a compact space, is also countably compact. This is because the Cantor set is uncountable, and therefore every open cover of the Cantor set is uncountable, and therefore has a finite subcover.

In Relation to Other Topological Properties

Countably compact spaces are related to several other topological properties.

Compactness

As mentioned above, every compact space is countably compact. However, the converse is not true. There exist countably compact spaces that are not compact. An example of such a space is the first uncountable ordinal with the order topology.

Lindelöf Property

A topological space is Lindelöf if every open cover of the space has a countable subcover. Every countably compact space is Lindelöf, but the converse is not true. An example of a Lindelöf space that is not countably compact is the space of rational numbers with the standard topology.

See Also

• Compact space
• Sequentially compact space
• Lindelöf space

A close-up of a mathematical diagram illustrating the concept of countably compact spaces.

A close-up of a mathematical diagram illustrating the concept of countably compact spaces.