The second axiom of countability is the concept of a general topology . A topological space satisfies the second axiom of countability if it has a countable base .
The fulfillment of this axiom (the presence of a countable base of the topology) significantly affects the fundamental properties of spaces. For example, regular topological spaces with a countable base are normal and, moreover, metrizable. In the case of compact Hausdorff spaces, the reverse is also true - the metrizability implies the presence of a countable base of topology.
Examples
The following topological spaces satisfy the second axiom of countability:
- Compact metric spaces
- Euclidean and any of their subspaces
- Finite discrete space
Properties
- The second axiom of countability follows from the second axiom of countability .
- Separability follows from the second axiom of countability.
- For metric spaces, the second axiom of countability is equivalent to separability .
See also
- The first axiom of countability