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
Separation type
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	dissatisfies: regular space, completely regular space, normal space
Compactness type
compact space	Yes	any space with a cofinite topology is compact
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
Dense subsets
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
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.
Miscellaneous
Toronto space	Yes