Countable space with cofinite topology
This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces
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|
|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|
|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|
|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|
|separable space||Yes||the space is countable, so obviously it has a countable dense subset|
|second-countable space||Yes||in fact, the collection of all open subsets is countable because the number of cofinite subsets is countable|
|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.|
|resolvable space||Yes||take a partition into two countably infinite subsets. Both are dense.|