\begin{definition}[Definition:Simplified Arens Square]
Let $A$ be the set of points in the interior of the unit square:
:$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \R} = \openint 0 1^2$
:rightthumb300pxSimplified Arens square with example local bases marked
Let $S$ be the set defined as:
:$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} }$
Let $\BB$ be the basis for a topology generated on $S$ be defined by granting:
:to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square;
:to the other points of $S$ the following local bases:
{{begin-eqn}}
{{eqn | l = \map {U_n} {0, 0}
      | o = :=
      | r = \set {\tuple {x, y}: 0 < x < \dfrac 1 2, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} }
}}
{{eqn | l = \map {U_m} {1, 0}
      | o = :=
      | r = \set {\tuple {x, y}: \dfrac 1 2 < x < 1, 0 < y < \dfrac 1 m} \cup \set {\tuple {1, 0} }
}}
{{end-eqn}}
Let $\tau$ be the topology generated from $\BB$.
$\struct {S, \tau}$ is referred to as the '''simplified Arens square'''.
\end{definition}