Definition:Alexandroff Square

Definition

 * Alexandroff-Square.png

Let $X$ be the closed unit square $\closedint 0 1 \times \closedint 0 1$.

Let $\Delta$ denote the diagonal of $X$:
 * $\Delta = \set {\tuple {x, x}: x \in \closedint 0 1}$

Let the topology $\tau$ be defined by means of the neighborhood basis defined as:


 * all $p = \tuple {s, t}$ such that $s \ne t$ with the vertical line segment:
 * $\map {N_\epsilon} {s, t} := \set {\tuple {s, y} \in X \setminus \Delta: \size {t - y} < \epsilon}$


 * all $\tuple {s, s} \in \Delta$ with the intersection of $X$ with horizontal regions excluding a finite number of vertical line segments:
 * $\map {M_\epsilon} {s, s} := \set {\tuple {x, y} \in X: \size {y - s} < \epsilon, x \ne x_0, x_1, \ldots, x_n}$

The topological space $T = \struct {S, \tau}$ is referred to as the Alexandroff square.

Also see

 * Alexandroff Square Topology is Topology