Definition:Dieudonné Plank

Definition

Let $\omega$ be the first transfinite ordinal.

Let $\Omega$ be the first uncountable ordinal.

Let $S = \paren {\closedint 0 \Omega \times \closedint 0 \omega} \setminus \set {\tuple {\Omega, \omega} }$ be the underlying set of the deleted Tychonoff plank.

Let the topology $\tau$ be generated by declaring as open:

each point of $\hointr 0 \Omega \times \hointr 0 \omega$

together with the sets defined as:

\(\ds \map {U_\alpha} \beta\)

\(=\)

\(\ds \set {\tuple {\beta, \gamma}: \alpha < \gamma \le \omega}\)

\(\ds \map {V_\alpha} \beta\)

\(=\)

\(\ds \set {\tuple {\gamma, \beta}: \alpha < \gamma \le \Omega}\)

The topological space $\struct {S, \tau}$ is known as the Dieudonné plank.

Also see

• Dieudonné Plank is Topology

• Results about the Dieudonné plank can be found here.

Source of Name

This entry was named for Jean Alexandre Eugène Dieudonné.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $89$. Dieudonné Plank