Example

Any discrete space is Hausdorff. The rational line $Q$ is Hausdorff, for if $x, y$ are two rational numbers such that $x < y$ then there is a rational number $z$ such that $x < z < y$, and the neighbourhoods ]$\leftarrow, z[$ of $x$ and ]$z, \rightarrow[$ of $y$ do not intersect.

A set $X$ having at least two points and carrying the coarsest topology (§ 2, no. 2) is not a Hausdorff space.

Let $f : X \to Y$ be a mapping of a set $X$ into a Hausdorff space $Y$; then it follows immediately from Proposition 1 that $f$ can have at most one limit with respect to a filter $\mathfrak{F}$ on $X$, and that if $f$ has $y$ as a limit with respect to $\mathfrak{F}$, then $y$ is the only cluster point of $f$ with respect to $\mathfrak{F}$.