Definition:Arens Square

Definition
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 $\dfrac 1 2$.


 * Arens-square.png

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:

Let $\tau$ be the topology generated from $\BB$.

$\struct {S, \tau}$ is referred to as the Arens square.

Also see

 * Arens Square is Topology


 * Definition:Simplified Arens Square