I am illiterate in algebra/set theory, so I hope you will forgive my ignorance.
It is known that the symmetric group of a finite set is generated by elementary permutations. More precisely, $Sym\{1,...,n\}$ is generated by $(1,2),...,(n-1,n)$. Now to what extend does this generalize to infinite sets?
a. For instance, is $Sym(\mathbb{N})$ generated by $(1,2),...$ ? I know that the group is of cardinality at least continuum (take Riemann's theorem on conditionally converging sequences). But so is that of the set of natural sequences, so I have some hope for an injective map. If not, what would be a good generalization of the construction that works?
b. What about $Sym(\mathbb{R})$? I am not even sure what 'generate' would mean in this case, a composition of a continuum number of elements? Say, a function $\mathbb{R}\mapsto G$, where G is the set of generators. Can $Sym(\mathbb{R})$ be injectively mapped into the set of maps $\mathbb{R}\mapsto G$ for some meaningful $G\subset Sym(\mathbb{R})$? Can $G$ be
$$ G=\{P_{x,y}| x\in\mathbb{R}, y\in\mathbb{R}_+\},\quad P_{x,y}(t)=x\chi_{\{x-y\}}(t)+(x-y)\chi_{\{x\}}(t)+t(1-\chi_{\{x-y\}}(t)-\chi_{\{x\}}(t)), $$
where $\chi_A$ is the characteristic function of $A\subset\mathbb{R}$. In other words, $P_{x,y}$ just permutes $x$ and $x-y$.
If not, what would be a working alternative? Thank you.