Definition:Smirnov's Deleted Sequence Topology

Definition

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $A$ denote the set defined as:

$A := \set {\dfrac 1 n: n \in \Z_{>0} }$

Let $\tau$ be the topology defined as:

$\tau = \set {H: \exists U \in \tau_d, B \subseteq A: H = U \setminus B}$

That is, $\tau$ consists of the open sets of $\struct {\R, \tau_d}$ which have had any number of the set of the reciprocals of the positive integers removed.

$\tau$ is then referred to as Smirnov's deleted sequence topology on $\R$.

Also known as

$\tau$ can also be seen referred to as just the Smirnov topology.

Some sources call it the $K$-topology, after a conventional definition of $K$ as being the set $\set {\dfrac 1 n: n \in \Z_{>0} }$.

Also see

• Smirnov's Deleted Sequence Topology is Topology

• Results about Smirnov's deleted sequence topology can be found here.

Source of Name

This entry was named for Yurii Mikhailovich Smirnov.

Historical Note

Smirnov's deleted sequence topology appears first to have been discussed by Yurii Mikhailovich Smirnov in his $1951$ paper in connection with his metrization theorem.

Sources

• 1951: Ю.М. Смирнов: О метризации топологических пространств ("On metrization of topological spaces") (Успехи Математических Наук Vol. 6: pp. 100 – 111)
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $64$. Smirnov's Deleted Sequence Topology