Countable space with cofinite topology: Difference between revisions
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* Its underlying set is an infinite countable set. | * Its underlying set is an infinite countable set. | ||
* The topology on it is a [[cofinite topology]]. | * 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== | |||
{| class="sortable" border="1" | |||
! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied | |||
|- | |||
| [[satisfies property::T1 space]] || Yes || points are closed by definition || satisfies: [[satisfies property::Kolmogorov space]] | |||
|- | |||
| [[satisfies property::sober space]] || Yes || we can see this from its algebraic variety interpretation, or directly || | |||
|- | |||
| [[dissatisfies property::Hausdorff space]] || No || any two non-empty open subsets intersect, so the space is far from Hausdorff || [[dissatisfies property::regular space]], [[dissatisfies property::completely regular space]], [[dissatisfies property::normal space]] | |||
|- | |||
| [[satisfies property::compact space]] || Yes || any space with a cofinite topology is compact || | |||
|- | |||
| [[satisfies property::connected space]] || Yes || any two non-empty open subsets intersect, so the space must be connected || | |||
|- | |||
| [[satisfies property::locally connected space]] || Yes || || | |||
|- | |||
| [[dissatisfies property::path-connected space]] || No || [[countable space with cofinite topology is not path-connected]] || | |||
|} | |||
Revision as of 18:15, 26 January 2012
This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces
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 |
|---|---|---|---|
| T1 space | Yes | points are closed by definition | satisfies: Kolmogorov space |
| sober space | Yes | we can see this from its algebraic variety interpretation, or directly | |
| Hausdorff space | No | any two non-empty open subsets intersect, so the space is far from Hausdorff | regular space, completely regular space, normal space |
| compact space | Yes | any space with a cofinite topology is compact | |
| 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 |