Question

If \(X\) is an infinite set, the cofinite topology on \(X\) is \(T_{1}\) but not \(T_{2}\), and is first countable iff \(X\) is countable.

Short answer

Cofinite topology on an infinite set is \(T_{1}\) but not \(T_{2}\), and first countable iff \(X\) is countable.

Step-by-step solution

Understanding Cofinite Topology

The cofinite topology on a set \(X\) consists of all subsets of \(X\) whose complements are either finite or \(X\) itself. It is important to understand how this topology behaves in terms of separation axioms \(T_{1}\) and \(T_{2}\), and countability properties.

Verifying \(T_{1}\) property

In the cofinite topology, for any two distinct points \(x\) and \(y\) in \(X\), the set \(X \setminus \{y\}\) is open and contains \(x\) but not \(y\). This means that for every pair of distinct points, we can find an open set containing one point and not the other, satisfying the \(T_{1}\) separation axiom.

Checking \(T_{2}\) property (Hausdorff)

For a topology to be \(T_{2}\), or Hausdorff, we should find disjoint open sets containing two distinct points. In a cofinite topology, any two open sets will have non-empty intersection since the intersection of two complements of finite sets is not guaranteed to be finite. Hence, the cofinite topology is not \(T_{2}\).

Analyzing First Countability

A space is first countable if each point has a countable local base. In the cofinite topology, any local base around a point must include open sets that contain all but finitely many points, which becomes unmanageable if \(X\) is uncountable since the number of finite sets is countable. Thus, the cofinite topology is first countable only if \(X\) is countable.

Conclusion

The cofinite topology on an infinite set \(X\) satisfies \(T_{1}\) but not \(T_{2}\) due to the definition of open sets intersecting non-trivially, and it is first countable if and only if \(X\) is countable because of the limitations in forming countable local bases.

Key concepts

Cofinite Topology

The cofinite topology on a set \(X\) is defined by its unique approach to open sets. It consists of all subsets whose complements in \(X\) are either finite or equal to \(X\) itself. This means that only finite sets have a complement that qualifies as an open set in this topology. A simple way to remind yourself of this is to think "cofinite means complements are finite."

This topology displays interesting behaviors that differentiate it from other topologies. Firstly, it changes our usual sense of 'closeness and separation' of points in a topological space. Despite its seemingly restrictive nature, the cofinite topology exists widely in the realm of mathematical topology as a basic and instructive example, especially when discussing separation properties and countability.

Separation Axioms

Separation axioms are critical in understanding how topologies distinguish between points and sets. In the context of the cofinite topology, these axioms help us recognize what kind of separation is possible.

• The \(T_1\) Axiom: A topological space is \(T_1\) if, for any pair of distinct points, each point has a neighborhood not containing the other. Cofinite topology is \(T_1\) since, if you pick any distinct points \(x\) and \(y\), you can cover all but \(y\) with the open set \(X \setminus \{y\}\), excluding \(y\) and including \(x\).
• The \(T_2\) Axiom (Hausdorff): Normally, a topology is \(T_2\) if every pair of distinct points has their own neighborhoods that do not overlap. However, the cofinite topology on an infinite set \(X\) is not \(T_2\), because when you attempt to find two open sets around two distinct points, those will inevitably intersect due to their nature of having complements that are finite at best.

Understanding these axioms within the cofinite topology framework accentuates the differences between \(T_1\) and \(T_2\) and sheds light on how some topologies manage point separation effectively, while others do not.

Countability Properties

Countability properties inform us about the structure and limitations of a topology relating to 'size' at a microscopic level. In cofinite topology, countability plays a key role in determining the complexity or simplicity of local neighborhoods.

First countability means that each point in the space has a base (a set of neighborhoods) that is countable. In a cofinite topology, for an infinite set \(X\), having a countable base is feasible only if \(X\) itself is countable. This is because:

• Any local base must cover all but a finite number of points, which makes it tricky to maintain countability if \(X\) is uncountable.
• Suppose we have an uncountable \(X\), then it is impossible to find a base of neighborhoods for any single point that remains countable, simply because the requirement to exclude only finitely many points consistently is overwhelming.

This distinct characteristic of the cofinite topology demonstrates the dependence of first countability on the cardinality of \(X\), and highlights why such topologies unfold numerous interesting discussions in advanced mathematical studies.