Definition 7

Let $(x_n)_{n \in \mathbf{N}}$ be an infinite sequence of elements of a set $X$. The elementary filter associated with the sequence $(x_n)$ is the filter generated by the image of the Fréchet filter (no. 1) by the mapping $n \to x_n$ of $\mathbf{N}$ into $X$.

It comes to the same thing to say that the elementary filter associated with $(x_n)$ is the set of subsets $M$ of $X$ such that $x_n \in M$ except for a finite number of values of $n$. If $S_n$ denotes the set of all $x_p$ such that $p \geq n$, then the sets $S_n$ form a base of the elementary filter associated with the sequence $(x_n)$.

The elementary filter associated with an infinite subsequence of a sequence $(x_n)$ is finer than the elementary filter associated with $(x_n)$ (cf. Exercise 15).

By definition, every elementary filter has a countable base. Conversely: