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

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