Definition:Thomas's Plank/Definition 2

Definition

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

• Equivalence of Definitions of Thomas's Plank

• 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.) ... (previous): Part $\text {II}$: Counterexamples: $93$. Thomas's Plank