Fréchet filter

In mathematics, the Fréchet filter, also called the cofinite filter, on a set ${\displaystyle X}$ is a certain collection of subsets of ${\displaystyle X}$ (that is, it is a particular subset of the power set of ${\displaystyle X}$). A subset ${\displaystyle F}$ of ${\displaystyle X}$ belongs to the Fréchet filter if and only if the complement of ${\displaystyle F}$ in ${\displaystyle X}$ is finite. Any such set ${\displaystyle F}$ is said to be cofinite in ${\displaystyle X}$, which is why it is alternatively called the cofinite filter on ${\displaystyle X}$.

The Fréchet filter is of interest in topology, where filters originated, and relates to order and lattice theory because a set's power set is a partially ordered set under set inclusion (more specifically, it forms a lattice). The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology.