# Prove that $A \mapsto \chi_A$ defines bijection between $\beta(\gamma)$ and set of function ${(0,1)}^{\omega}$

I was given the following task

Let $$\Omega$$ - be some set (finire or infinite)
Assign to each subset A $$\subset\Omega$$ function $$\chi_A : \Omega \to {(0,1)}$$
Where $$\chi_A$$ is defined as $$\chi_A(x) = 1$$ if $$x \in A$$ and $$\chi_A(x) = 0$$ if $$x \notin A$$

I need to prove that

$$A \mapsto \chi_A$$ defines bijection between $$\cal P(\Omega)$$ (all the subsets of set $$\Omega$$) and the set of functions $${(0,1)}^{\omega}$$

In this problem, I do not understand the following :

1. What does it mean that some function defines a bijection between set A and set B.
2. What is the set of functions $${(0,1)}^{\omega}$$
• What is $\beta(\Omega)$? – J.-E. Pin Sep 16 at 6:19
• $\beta(\Omega)$ all the subsets of set $\Omega$ – Rustem Sadykov Sep 16 at 6:22
• The standard notation for the set of all subsets of $\Omega$ is ${\cal P}(\Omega)$. – J.-E. Pin Sep 16 at 6:25

There are several notation issues in your question. The right statement of the question should be

Show that the function $$\chi:A \to \chi_A$$ defines a bijection from $${\cal P}(\Omega)$$ to $$\{0,1\}^\Omega$$.

The set $$\{0,1\}^\Omega$$ is the set of all functions from $$\Omega$$ to $$\{0,1\}$$. In particular, for each subset $$A$$ of $$\Omega$$, $$\chi_A$$ belongs to $$\{0,1\}^\Omega$$.