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