# Deleted Integer Topology is Second-Countable

## Theorem

Let $S = \R_{\ge 0} \setminus \Z$.

Let $\tau$ be the deleted integer topology on $S$.

Then the topological space $T = \left({S, \tau}\right)$ is second-countable.

### Corollary 1

The topological space $T = \left({S, \tau}\right)$ is first-countable.

### Corollary 2

The topological space $T = \struct {S, \tau}$ is separable.

### Corollary 3

The topological space $T = \left({S, \tau}\right)$ is Lindelöf.

## Proof

Let $\Z_{> 0}$ be understood as the set of strictly positive integers:

- $\Z_{> 0} = \left\{{x \in \Z: x > 0}\right\} = \left\{{1, 2, 3, \ldots}\right\}$

From Basis for Partition Topology, the set:

- $\mathcal B = \left\{{\left({n - 1 \,.\,.\, n}\right): n \in Z_{> 0}}\right\}$

is a basis for $T$.

There is an obvious one-to-one correspondence $\phi: \Z_{> 0} \leftrightarrow \mathcal B$ between $\Z_{> 0}$ and $\mathcal B$:

- $\forall x \in \Z_{> 0}: \phi \left({x}\right) = \left({x - 1 \,.\,.\, x}\right)$

But $\Z_{> 0} \subseteq \Z$ and Integers are Countably Infinite.

So from Subset of Countably Infinite Set is Countable, $\Z_{> 0}$ is countable.

Thus $\mathcal B$ is also countable by definition of countability.

So we have that $T$ has a countable basis, and so is second-countable by definition.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 7: \ 5$