Definition:Alexandroff Plank

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Let $\Omega$ be the first uncountable ordinal.

Let the closed ordinal space $\closedint 0 \Omega$ be given the interval topology.

Let the closed real interval $\closedint {-1} 1$ be given the interval topology.

Let $\struct {S, \tau}$ be the product space of $\closedint 0 \Omega$ with $\closedint {-1} 1$

Consider the point $p := \tuple {\Omega, 0} \in S$

Let $\sigma$ be the expansion of $\tau$ generated by adding to $\tau$ the sets of the form:

$\map U {\alpha, n} := \set p \cup \hointl \alpha \Omega \times \openint 0 {\dfrac 1 n}$

Then the topological space $T = \struct {S, \sigma}$ is known as the Alexandroff plank.

Also see

  • Results about the Alexandroff plank can be found here.

Source of Name

This entry was named for Pavel Sergeyevich Alexandrov.