Topology,Number, Boolean Ring

Consider the factor set(not multiset) of some natural number like $m=p_1{p}_2{p}_3...p_n$$m=p_1p_2p_3...p_n$

Denote that all the natural numbers can be written as $p_i{p}_j{p}_k...p_{\ast }$$p_ip_jp_k...p_*$ as ${\mathbb{N}}^{\ast }$$\mathbb N^*$

$\mathrm{factor}\{m\}=\{p_1,p_2,p_3,...,p_n\}$$\mathrm{factor}\{m\}=\{p_1,p_2,p_3,...,p_n\}$

We know that $\mathrm{factor}\{1\}=\mathrm{\varnothing }$$\mathrm{factor}\{1\}=\empty$, and some natural number like $m$$m$, their factor set is a subset of ${\mathbb{N}}^{+}$$\mathbb N^+$

and $gcd(a,b)=\text{factor}(a)\cap \text{factor}(b),\text{factor}(a)\cup \text{factor}(b)$$\gcd(a,b)=\text{factor}(a)\cap \text{factor}(b),\text{factor}(a)\cup \text{factor}(b)$

and it is closed for $gcd(a_i{)}_{i=1}^{\mathrm{\infty }}$$\gcd(a_i)_{i=1}^{\infty}$, $\mathrm{lcm}(a_i{)}_{i=1}^n$$\mathrm{lcm}(a_i)_{i=1}^n$

So, define a topology ${\tau }_{\ast }$$\tau_*$ as follows

$†.\mathrm{\varnothing },{\mathbb{N}}^{\ast },\text{factor}(a),a\in {\mathbb{N}}^{\ast }$$\dagger.\empty,\mathbb N^*,\text{factor}(a),a\in \mathbb N^*$ is closed set

Observe that for $S,T\in {\tau }_{\ast },S-T\in {\tau }_{\ast }$$S,T\in\tau_{*},S-T\in\tau_*{}$

Thus it is also Closed for $S\ominus T:=S\cup T-S\cap T$$S\ominus T:= S\cup T-S\cap T$

Thus it can be a Boolean Ring