Definition:Alexandroff Square
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Definition
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
- Results about the Alexandroff square can be found here.
Source of Name
This entry was named for Pavel Sergeyevich Alexandrov.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $101$. Alexandroff Topology