\begin{definition}[Definition:Minimal Hausdorff Topology]
:thumbright420pxExamples of [[Definition:Neighborhood Basisneighborhood bases of $a$ and $-a$]]
Let $\omega$ be the first transfinite ordinal.
Let $\struct {A, \tau_1}$ denote the totally ordered set defined as:
:$A := \set {1, 2, 3, \ldots, \omega, \ldots, -3, -2, -1}$
with the interval topology $\tau_1$.
Let $\struct {\Z_{>0}, \tau_2}$ denote the (strictly) positive integers with the discrete topology $\tau_2$.
Let $S$ be defined as:
:$S = A \times \Z_{>0} \cup \set {a, -a}$
where $a$ and $-a$ are new elements of $S$.
Let $\tau$ be the topology defined on $S$ as:
:the product topology on $\struct {A, \tau_1} \times \struct {\Z_{>0}, \tau_2}$
together with neighborhood bases of $a$ and $-a$ defined as:
{{begin-eqn}}
{{eqn | l = \map { {M_n}^+} a
      | o = :=
      | r = \set a \cup \set {\tuple {i, j}: i < \omega, j > n}
}}
{{eqn | l = \map { {M_n}^-} {-a}
      | o = :=
      | r = \set {-a} \cup \set {\tuple {i, j}: i > \omega, j > n}
}}
{{end-eqn}}
$\tau$ is referred to as the '''minimal Hausdorff topology'''.
The topological space $T = \struct {S, \tau}$ is referred to as the '''minimal Hausdorff space'''.
\end{definition}