# example of uncountable family of subsets of a countable set with finite intersections

We wish to give an answer to the following:

Problem. Assume, that $X$ is a countable set. Is there a family ${\{{X}_{i}\}}_{i\in I}$ of subsets of $X$ such that $I$ is an uncountable set, but for any $i\ne j\in I$ the intersection^{} ${X}_{i}\cap {X}_{j}$ is finite?

Example. Let $x\in [1,2)$ be a real number. Express $x$ using digits

$$x=1.{x}_{1}{x}_{2}{x}_{3}{x}_{4}\mathrm{\cdots}=1+\sum _{i=1}^{\mathrm{\infty}}{x}_{i}\cdot {10}^{-i}$$ |

where each ${x}_{i}\in \{0,1,2,3,4,5,6,7,8,9\}$. With $x$ we associate the following natural numbers^{}

$${\beta}_{n}(x)=1{x}_{1}{x}_{2}{x}_{3}\mathrm{\cdots}{x}_{n-1}{x}_{n}={10}^{n+1}+\sum _{i=1}^{n}{x}_{i}\cdot {10}^{n-i+1}.$$ |

Now define $A:[1,2)\to \mathrm{P}(\mathbb{N})$ (here $\mathrm{P}(X)$ stands for ,,the power set^{} of $X$”) by

$$A(x)=\{{\beta}_{1}(x),{\beta}_{2}(x),{\beta}_{3}(x),\mathrm{\dots}\}.$$ |

$A$ is injective^{}. Indeed, note that for any $x,y\in [1,2)$ if ${\beta}_{i}(x)={\beta}_{j}(y)$, then $i=j$ (this is because equal $\beta $ numbers have equal ,,length” and this is because each $\beta $ has $1$ at the begining, zeros are not the problem). Therefore, if $A(x)=A(y)$ for some $x,y$, then it follows, that ${\beta}_{i}(x)={\beta}_{i}(y)$ for each $i$, but this implies that corresponding digits of $x$ and $y$ are equal. Thus $x=y$.

This shows, that ${\{A(x)\}}_{x\in [1,2)}$ is an uncountable family of subsets of $\mathbb{N}$. Now in order to prove that $A(x)\cap A(y)$ is finite whenever $x\ne y$ it is enough to show that we can uniquely reconstruct $x$ from any infinite^{} sequence of numbers from $A(x)$. This can be proved by using similar techniques as before and we leave it as a simple exercise.

Title | example of uncountable family of subsets of a countable set with finite intersections |
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Canonical name | ExampleOfUncountableFamilyOfSubsetsOfACountableSetWithFiniteIntersections |

Date of creation | 2013-03-22 19:16:29 |

Last modified on | 2013-03-22 19:16:29 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Example |

Classification | msc 03E10 |