The Order Axioms of ℝ Notes

Real analysis often works with intervals, which are sets of real numbers defined by inequalities.

Types of Intervals:

• Open interval:

  $(a, b) = \{ x \in ℝ \mid a < x < b \}$

  (a,b)={x∈R∣a<x<b}

• Closed interval:

  $[a, b] = \{ x \in ℝ \mid a \leq x \leq b \}$

  [a,b]={x∈R∣a≤x≤b}

• Half-open:

  $(a, b], [a, b)$

  (a,b],[a,b)

• Infinite:

  $(-\infty, b), (a, \infty), (-\infty, \infty)$

  (−∞,b),(a,∞),(−∞,∞)

These play an essential role in defining limits, continuity, and compactness.