The Cofinite Topology

The Cofinite Topology

Recall from the Topological Spaces page that a set $X$ an a collection $\tau$ of subsets of $X$ together denoted $(X, \tau)$ is called a topological space if:

• $\emptyset \in \tau$ and $X \in \tau$, i.e., the empty set and the whole set are contained in $\tau$.
• If $U_i \in \tau$ for all $i \in I$ where $I$ is some index set then $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$, i.e., for any arbitrary collection of subsets from $\tau$, their union is contained in $\tau$.
• If $U_1, U_2, ..., U_n \in \tau$ then $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$, i.e., for any finite collection of subsets from $\tau$, their intersection is contained in $\tau$.

We will now look at a rather interested topology known as the cofinite topology.

 Definition: If $X$ be a nonempty set, then the Cofinite Topology of $X$ is the collection of subsets $\tau = \{ U \subseteq X : U = \emptyset \: \mathrm{or} \: U^c \: \mathrm{is \: finite} \}$.

Another term for the cofinite topology is the "Finite Complement Topology".

Let's verify that $(X, \tau)$ is a topological space.

For the first condition, we clearly see that $\emptyset \in \tau = \{ U \subseteq X : U = \emptyset \: \mathrm{or} \: U^c \: \mathrm{is \: finite} \}$. Furthermore, $X^c = \emptyset$ (noting that the universal set in this instance is $X$ itself), so $X \in \tau$.

For the second condition, let $\{ U_i \}_{i \in I}$ be a collection of subsets of $\tau$ for some index set $I$. By the generalized De Morgan's Laws we have that:

(1)
\begin{align} \quad \left ( \bigcup_{i \in I} U_i \right )^c = \bigcap_{i \in I} U_i^c \end{align}

Suppose that $U_i = \emptyset$ for some $i \in I$. Then $\displaystyle{\bigcap_{i \in I} U_i^c = \emptyset \in \tau}$. Now suppose that $U_i \neq \emptyset$ for all $i \in I$. Then $U_i^c$ is finite for all $i \in I$ we must have that $\displaystyle{\bigcap_{i \in I} U_i^c}$ is finite. Hence $\displaystyle{\left ( \bigcup_{i \in I} U_i \right )^c}$ is finite, so $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$.

For the third condition, let $U_1, U_2, ..., U_n \in \tau$. Then by applying the generalized De Morgan's laws once again we see that:

(2)
\begin{align} \quad \left ( \bigcap_{i=1}^{n} U_i \right )^c = \bigcup_{i=1}^{n} U_i^c \end{align}

Suppose that $U_i = \emptyset$ for some $i \in \{1, 2, ..., n \}$. Then $\displaystyle{\bigcup_{i=1}^{n} U_i^c = X \in \tau}$. Now suppose that $U_i \neq \emptyset$ for all $i \in \{1, 2, ..., n \}$. Then $U_i^c$ is finite for all $i \in \{1, 2, ..., n \}$ we must have that $\displaystyle{\bigcup_{i=1}^{n} U_i^c}$ is finite. Hence $\displaystyle{\left ( \bigcap_{i=1}^{n} U_i \right )^c}$ is finite, so $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$.

Therefore $(X, \tau )$ is a topological space.

 Proposition 1: If $X$ is a finite set then the cofinite topology on $X$ is the discrete topology.
• Proof: Suppose that $X$ is a finite set. Then every subset $U \subseteq X$ is finite, and $U^c = X \setminus U$ is finite. Therefore, for every subset $U \subseteq X$ we have that $U \in \tau$, so $\tau$ contains all subsets of $X$, i.e., $\tau = \mathcal P (X)$ so the cofinite topology on $X$ is the discrete topology. $\blacksquare$