# Countable space with cofinite topology

## 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.