Odd-Even Topology is Second-Countable

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.


Then $T$ is second-countable.


Corollary 1

$T$ is first-countable.


Corollary 2

$T$ is separable.


Corollary 3

$T$ is Lindelöf.


Proof

From Basis for Partition Topology, the set:

$\BB := \set {\set {2 k - 1, 2 k}: k \in \Z, k > 0}$

is a basis for $T$.

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

$\forall x \in \Z_{>0}: \map \phi x = \set {2 x - 1, 2 x}$

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

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

Thus $\BB$ 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