Definition:Thomas's Plank

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Definition

Definition 1

Let $L_n$ be lines embedded in the Cartesian plane $\R^2$ defined as:

$\forall n \in \N: L_n = \begin {cases} \set {\tuple {x, 0}: x \in \openint 0 1} & : n = 0 \\ \set {\tuple {x, \dfrac 1 n}: x \in \hointr 0 1} & : n > 0 \end {cases}$

Let $S = \ds \bigcup {n \mathop \in \N}$.

Let a topology $\tau$ be applied to $S$, defined as follows:

For $n \ge 1$, each point of $L_n$ except for $\tuple {0, \dfrac 1 n}$ is open.
neighborhood bases of $\tuple {0, \dfrac 1 n}$ are subsets of $L_n$ with finite complements.
neighborhood bases of $\tuple {x, 0}$ are the sets $\map {U_i} {x, 0}$ defined as:
$\map {U_i} {x, 0} := \set {\tuple{x, \dfrac 1 n}: n > i}$


Thomas's plank is the topological space $\struct {S, \tau}$.


Definition 2

Let $L := \openint 0 1$ denote the open unit interval.

Let $S_1 := L \cup \set p$ denote the Alexandroff extension of $L$.

Let $S_2 := \Z_{>0} \cup \set q$ denote the Alexandroff extension of the (strictly) positive integers $\Z_{>0}$.

Let $\struct {S, \tau} := \paren {S_1 \times S_2} \setminus \set {\tuple {p, q} }$ be the subspace of the product space $S_1 \times S_2$ with $\set {\tuple {p, q} }$ removed.

Thomas's plank is the topological space $\struct {S, \tau}$.


Also see

  • Results about Thomas's plank can be found here.


Source of Name

This entry was named for John David Thomas.


Sources