Definition:Thomas's Plank
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:
- 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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $93$. Thomas's Plank