Example of irreducible lattice

Question

I think I have proofed that $\hbox{Bor}([0,1])$ is an example of irreducible lattice. Please, tell me if I am wrong, and if I am, then show me where exactly. Here is what I have done.

1. $|\hbox{Bor}([0,1])| = 2^{\aleph_0}$.

2. If there exists product $A = \prod_{t\in T}A_t$ isomorphic to $\hbox{Bor}([0,1])$, then (let $\phi$ be the isomorphism) for every $x\in [0,1]$ $|\hbox{supp}(\phi(\lbrace x\rbrace))| = 1$, where $\hbox{supp}(f) = \lbrace t\in T| f(t)\neq \phi(\emptyset)(t)\rbrace$.

Because of 2, $|T| \ge |\lbrace \lbrace x\rbrace\colon x\in [0,1]\rbrace| = 2^{\aleph_0}$. WLoG we can assume, that for every $t$ $|A_t|\ge 2$. So $|A|\ge 2^{2^{\aleph_0}}$. Contradiction with 1, the proof is over.

Is it ok?

I have another question. If the topological space $T$ has cardinality $\kappa$, then is it true that $|\hbox{Bor}(T)| = \kappa$? If it is true, then one can construct arbitrary big irreducible lattice, right?

But I guess something is wrong here :(

Answer 1

Is it ok?

The question is whether this sketch of an argument correctly proves that the lattice of Borel subsets of $[0,1]$ is directly indecomposable. The answer is No, the error in the argument is the assumption that $x\neq y$ implies $\textrm{supp}(\phi(\{x\}))\neq \textrm{supp}(\phi(\{y\}))$.

The correct answer is: $\textrm{Bor}([0,1])$ IS directly decomposable. It is a Boolean lattice, and the only nontrivial directly indecomposable Boolean lattice is the 2-element chain.

For a direct argument, let $A$ be a nonempty proper Borel subset of $[0,1]$. E.g., take $A=[0,\frac{1}{2}]$. Let $A^c$ be the complement of $A$. The function $$\varphi\colon \textrm{Bor}([0,1])\to \textrm{Bor}(A)\times \textrm{Bor}(A^c)\colon X\mapsto (X\cap A,X\cap A^c)$$ is an isomorphism from $\textrm{Bor}([0,1])$ onto a nontrivial product of lattices.