# 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

- 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