Here is my previous question Investigating the relationship between $X$ and the inverse limit $\mathcal{L}$ of the tower.
And now I want to Find a way to uniquely determine a point $x_{\sigma} \in X,$ given an element $\sigma \in \mathcal{L}.$ This defines a function $f:\mathcal{L} \rightarrow X.$
Could anyone help me in doing so please?
As I said in my answer, we can define it by $\{x_\sigma\} = \bigcap_n \sigma_n$.
The map is 1-1 because if $\sigma \neq \sigma'$, then for some $n$ we have $\sigma_n \neq \sigma'_n$ and as these are from the same $\mathcal{A}_n$, which is a partition, $\sigma_n \cap \sigma'_n = \emptyset$ which implies $x_\sigma \neq x_{\sigma'}$.
The map is also onto, as each $x$ occurs in a unique $\sigma_n \in \mathcal{A}_n$ and then the $\sigma$, so defined, has $x_\sigma=x$ by definition.
The last function also gives the inverse $g: X \to \mathcal{L}$: $g(x)=\sigma$ defined by: $\sigma_n$ is the unique member of $\mathcal{A}_n$ that contains $x$.