What is vitali set?

The Vitali set is a classic example of a non-measurable set in mathematics. It was introduced by Giuseppe Vitali in 1905 to show that there exist sets that cannot be assigned a unique measure under the usual Lebesgue measure, which is a measure of the "size" of a set.

The Vitali set is constructed as follows: Take an interval [0,1] and assume that we can identify all its points with real numbers between 0 and 1. Choose a representative for each equivalence class under the relation of "difference is a rational number." This means that we have a collection of points, one from each equivalence class.

The Vitali set is then defined as the set of all representatives that we have chosen, i.e., the set of all points in [0,1] such that the difference between any two points is rational. It is easy to see that the Vitali set is uncountable, since there are uncountably many equivalence classes to choose representatives from.

The Vitali set has several interesting properties, including being unmeasurable and having the same cardinality as the real numbers. It is also an example of a non-Hausdorff space, since the points in the Vitali set cannot be separated by open sets.