\begin{definition}[Definition:Arens Square]
Let $A$ be the subset of the unit square defined as:
:$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \Q} \setminus \set {\tuple {i, j}: i = \dfrac 1 2}$
That is:
:$A := \openint 0 1^2 \cup \Q \times \Q \setminus \set {\tuple {\dfrac 1 2}, j: j \in \Q}$
That is, the set of rational points in the interior of the unit square from which are excluded the points whose $x$ coordinates are equal to $\dfrac 1 2$.
:rightthumb300pxArens square with example local bases marked
Let $S$ be the set defined as:
:$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} } \cup \set {\tuple {\dfrac 1 2, r \sqrt 2}: r \in \Q, 0 < r \sqrt 2 < 1}$
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 4, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} }
}}
{{eqn | l = \map {U_n} {1, 0}
      | o = :=
      | r = \set {\tuple {x, y}: \dfrac 3 4 < x < 1, 0 < y < \dfrac 1 n} \cup \set {\tuple {1, 0} }
}}
{{eqn | l = \map {U_n} {\tuple {\dfrac 1 2, r \sqrt 2} }
      | o = :=
      | r = \set {\tuple {x, y}: \dfrac 1 4 < x < \dfrac 3 4, \size {y - r \sqrt 2} < \dfrac 1 n}
}}
{{end-eqn}}
Let $\tau$ be the topology generated from $\BB$.
$\struct {S, \tau}$ is referred to as the '''Arens square'''.
\end{definition}