Definition:Thomas's Plank/Definition 1

Definition
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 $\struct {S, \tau}$.

Also see

 * Equivalence of Definitions of Thomas's Plank