Definition:Simplified Arens Square

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Definition

Let $A$ be the set of points in the interior of the unit square:

$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \R} = \openint 0 1^2$


Simplified 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:
\(\ds \map {U_n} {0, 0}\) \(:=\) \(\ds \set {\tuple {x, y}: 0 < x < \dfrac 1 2, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} }\)
\(\ds \map {U_m} {1, 0}\) \(:=\) \(\ds \set {\tuple {x, y}: \dfrac 1 2 < x < 1, 0 < y < \dfrac 1 m} \cup \set {\tuple {1, 0} }\)


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

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


Also see

  • Results about the simplified Arens square can be found here.


Source of Name

This entry was named for Richard Friederich Arens.


Sources