Countable space with cofinite topology
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Definition
This topological space is defined as follows:
- Its underlying set is an infinite countable set.
- The topology on it is a cofinite topology.
The topological space also arises as the Zariski topology for a countably infinite field as an algebraic variety over itself, or more generally, for any connected one-dimensional algebraic variety over a countably infinite field. Equivalently, it arises as the max-spectrum of the polynomial ring in one variable over a countably infinite field.
Topological space properties
| Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
|---|---|---|---|
| Separation type | |||
| T1 space | Yes | points are closed by definition | satisfies: Kolmogorov space |
| Hausdorff space | No | any two non-empty open subsets intersect, so the space is far from Hausdorff | dissatisfies: regular space, completely regular space, normal space |
| KC-space | No | all subsets are compact, but not necessarily closed | dissatisfies: Hausdorff space |
| locally Hausdorff space | No | any non-empty open subset is itself homeomorphic to the whole space, hence is not Hausdorff | dissatisfies: Hausdorff space |
| Compactness type | |||
| compact space | Yes | any space with a cofinite topology is compact | |
| Noetherian space | Yes | any proper closed subset is finite | satisfies: hereditarily compact space, compact space |
| Connectedness type | |||
| connected space | Yes | any two non-empty open subsets intersect, so the space must be connected | |
| locally connected space | Yes | ||
| path-connected space | No | countable space with cofinite topology is not path-connected | |
| irreducible space | Yes | cannot be expressed as a union of two proper closed subsets, because all such subsets are finite | |
| Countability type | |||
| separable space | Yes | the space is countable, so obviously it has a countable dense subset | |
| first-countable space | Yes | ||
| second-countable space | Yes | in fact, the collection of all open subsets is countable because the number of cofinite subsets is countable | |
| Miscellaneous | |||
| Baire space | No | the intersection of complements of points is empty, although each complement is open and dense and there are countably many of them. | |
| Toronto space | Yes | ||
| resolvable space | Yes | take a partition into two countably infinite subsets. Both are dense. |