A topology is meant to record which subsets of a space are visible as neighbourhoods, but the axioms alone allow spaces whose open sets are far from metric intuition. The cofinite topology is one of the first useful tests of that intuition. It asks what happens if the only closed sets small enough to name are finite sets, while every set with finite complement is declared open.

The surprising feature is that this topology is usually not discrete, not Hausdorff, and not metrizable, yet it still has strong compactness properties. It is therefore a compactness machine: it shows that compactness can come from having very few open sets, not only from boundedness or completeness.

[example: Two Infinite Open Sets Must Meet] Let $X$ be an infinite set, and suppose $U,V\subset X$ have finite complements. For each $x\in X$, \begin{align*} x\in X\setminus (U\cap V) \Longleftrightarrow x\notin U\cap V \Longleftrightarrow (x\notin U)\text{ or }(x\notin V). \end{align*} The last condition is equivalent to $x\in (X\setminus U)\cup (X\setminus V)$, so \begin{align*} X\setminus (U\cap V)=(X\setminus U)\cup (X\setminus V). \end{align*} Since $X\setminus U$ and $X\setminus V$ are finite, their union is finite. Hence $X\setminus (U\cap V)$ is finite, so $U\cap V$ has finite complement. If $U\cap V=\varnothing$, then \begin{align*} X\setminus (U\cap V)=X\setminus \varnothing=X, \end{align*} which would make $X$ finite, contradicting the assumption that $X$ is infinite. Therefore $U\cap V\ne\varnothing$. This example shows the basic failure of Hausdorff intuition: in the cofinite topology on an infinite set, two nonempty open sets cannot be separated from each other. [/example]

This single computation drives the whole chapter. Separation fails because open sets are too large; compactness holds because open covers cannot avoid covering all but finitely many points at once; convergence behaves strangely because to converge to a point, a sequence or net only needs eventually to enter every neighbourhood of that point, and each such neighbourhood contains a cofinite [open set](/page/Open%20Set). Here a net is indexed by a directed set $(D,\le)$: this means $\le$ is reflexive and transitive, and for any $d_1,d_2\in D$ there is some $e\in D$ with $d_1\le e$ and $d_2\le e$. This common-upper-bound condition is what makes phrases such as "eventually for all $d\ge d_0$" meaningful beyond ordinary sequences.

## Definition

Before defining the topology, we should isolate the question it answers. Suppose we want a topology on a set $X$ in which finite sets are the only closed sets forced by point data. In a [metric space](/page/Metric%20Space), points are closed, but many infinite sets are closed as well. The cofinite topology keeps the condition that points are closed while refusing to distinguish most infinite subsets from the whole space.

[definition: Cofinite Topology] Let $X$ be a set. The cofinite topology on $X$ is the collection \begin{align*} \tau_{\mathrm{cof}} = \{\varnothing\} \cup \{U \subset X : X \setminus U \text{ is finite}\}. \end{align*} The [topological space](/page/Topological%20Space) $(X,\tau_{\mathrm{cof}})$ is called a cofinite topological space. [/definition]

The definition says that a nonempty open set is allowed to miss only finitely many points. On a finite set this is no restriction at all, because every complement is finite. On an infinite set it is a severe restriction: there are many subsets of $X$ which are neither open nor closed.

The open-set definition is often easier to use for covers, while the closed-set form is better for separation and closure arguments. To use the topology effectively, we need to know exactly which subsets can be closed. The obstruction is that an infinite proper subset cannot be detected as closed: only finite exceptional sets, together with the whole space, survive as closed sets.

[quotetheorem:8843]

This theorem is the main translation device for the page. Whenever an argument mentions closure, density, or separation, it usually reduces to asking whether a finite set or the whole set is involved.

The finite case is worth separating at the beginning because it prevents many false contrasts later. In finite spaces the cofinite topology does not create an exotic example; it gives the most detailed possible topology.

[example: Finite Sets Become Discrete] Let $X=\{1,2,3\}$. To compute the cofinite topology, list the complements of all subsets of $X$: \begin{align*} X\setminus \varnothing=\{1,2,3\}. \end{align*} \begin{align*} X\setminus \{1\}=\{2,3\}, \qquad X\setminus \{2\}=\{1,3\}, \qquad X\setminus \{3\}=\{1,2\}. \end{align*} \begin{align*} X\setminus \{1,2\}=\{3\}, \qquad X\setminus \{1,3\}=\{2\}, \qquad X\setminus \{2,3\}=\{1\}. \end{align*} \begin{align*} X\setminus X=\varnothing. \end{align*} Each complement displayed is finite, because each is a subset of the finite set $X$. Therefore every subset of $X$ has finite complement, so every subset of $X$ is open in the cofinite topology: \begin{align*} \tau_{\mathrm{cof}}=\{\varnothing,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},X\}. \end{align*} Since the power set $\mathcal{P}(X)$ is exactly the collection of all subsets of $X$, we have \begin{align*} \tau_{\mathrm{cof}}=\mathcal{P}(X). \end{align*} Thus on this finite set the cofinite topology is the [discrete topology](/page/Discrete%20Topology), so the unusual behaviour of the cofinite topology depends on $X$ being infinite. [/example]

The infinite case is the real model. It is the smallest topology among those in which every finite set is closed, and it is a canonical example of a topology determined by a size condition rather than by distances.

## Open Sets and Closed Sets

### Large Open Sets

The cofinite topology is governed by a simple principle: a nonempty open set is large. This raises the basic consistency check for the definition: if two allowed open sets each miss only finitely many points, their intersection must still miss only finitely many points, while unions must not introduce any new obstruction. That finite-exception calculation is the reason the proposed collection is a topology.

[quotetheorem:8844]

The finite-intersection axiom is the only point where the word finite matters. Intersecting finitely many cofinite sets removes only finitely many points. Arbitrary intersections are not part of the topology axioms, and in fact arbitrary intersections of cofinite open sets need not be open.

To speak efficiently about neighbourhoods, we need the local version of openness. In a metric space a neighbourhood can be very small around a point. In a cofinite space every neighbourhood of a point contains almost every point of the space, so local information is already global information.