Finite complement topology

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More useful as an example than anything else

Definition

Let $X$ be an arbitrary set. There is a topology, $\mathcal{J}$, we can give $X$ called "the finite complement topology", such that $(X,\mathcal{J})$ is a topological space. It is defined as follows:

• $\mathcal{J}:\eq\left\{U\in\mathcal{P}(X)\ \big\vert\ U\eq\emptyset \vee\vert X-U\vert\in\mathbb{N}\right\}$, that is to say $U\in\mathcal{P}(X)$ is in $\mathcal{J}$ if $U\eq\emptyset$ or the complement of $U$ in $X$ has finite cardinality.

Hence the name "finite complement topology"

A topology must contain the empty set. Hence the first condition, note that $X-\emptyset\eq X$ which may not be finite! Thus $\emptyset$ might not otherwise be there.

See also

• Example:The real line with the finite complement topology is not Hausdorff