Category:Arens Square

This category contains results about 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 $\dfrac 1 2$.

Arens square with example local bases marked

Let $S$ be the set defined as:

$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} } \cup \set {\tuple {\dfrac 1 2, r \sqrt 2}: r \in \Q, 0 < r \sqrt 2 < 1}$

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:

$\ds \map {U_n} {0, 0}$

$:=$

$\ds \set {\tuple {x, y}: 0 < x < \dfrac 1 4, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} }$

$\ds \map {U_n} {1, 0}$

$:=$

$\ds \set {\tuple {x, y}: \dfrac 3 4 < x < 1, 0 < y < \dfrac 1 n} \cup \set {\tuple {1, 0} }$

$\ds \map {U_n} {\tuple {\dfrac 1 2, r \sqrt 2} }$

$:=$

$\ds \set {\tuple {x, y}: \dfrac 1 4 < x < \dfrac 3 4, \size {y - r \sqrt 2} < \dfrac 1 n}$

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

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

Subcategories

This category has only the following subcategory.