---- Let $X$ be any set; set $\tau_{f}:\{ U \subset X : X \cut U \text{ is finite or is all of } X \}$. > [!definition] Definition. ([[finite complement topology]]) > $\tau _f$ is a [[topological space|topology]] on $X$, called the **cofinite topology**. > [!justification] $X - X=\emptyset$ is clearly finite, and $X- \emptyset=X$. So $X$, $\emptyset \in \tau$. If $\{ U_{\alpha} \}$ is an indexed family of nonempty elements of $\tau_{f}$, then $X - \bigcup U_{\alpha}=\bigcap (X-U_{\alpha})$ and the latter is finite as an intersection of finite sets. If $U_{1},\dots,U_{n}$ are nonempty elements of $\tau_{f}$, to show $\bigcap_{i=1}^n U_{i} \in \tau_{f}$, we compute $X -\bigcap_{i=1}^n U_{i} =\bigcup_{i=1}^n(X-U_{i})$ which is finite as a finite union of finite sets. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```