A dense set is a mathematical concept in which every point in a space is either part of the set or is arbitrarily close to a point in the set. This means that between any two points in the space, no matter how close they are, there is always at least one point from the dense set. For example, the set of rational numbers ℚ is dense in the set of real numbers ℝ because between any two real numbers, you can find a rational number. Dense sets are important in various areas of mathematics, including analysis and topology. They help in understanding the structure of different number systems and the continuity of functions. Another example of a dense set is the set of irrational numbers ℝ \ ℚ, which is also dense in the real numbers. This property allows mathematicians to explore limits and convergence in mathematical analysis.