Order on the reals

Depends on

• The real numbers
• Order on the rationals

Used by

• The Cauchy-sequence reals have the least-upper-bound property Corollary
• Contractive sequence: |xₖ₊₂ - xₖ₊₁| ≤ c |xₖ₊₁ - xₖ| for a fixed 0 < c < 1 Definition
• Divergence to +∞ and to -∞ Definition
• Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length Definition
• Limits and Cauchy sequences of reals Definition
• Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric Definition
• Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of ℝ, with the dictionary to monotone sequences Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• Open ball, closed ball and sphere in a metric space Definition
• Open subset of ℝ (every point has a neighbourhood inside it), closed subset (complement open), and clopen Definition
• Sequences of reals: bounded, eventually, frequently, tails, subsequences Definition
• The closed long ray ω₁ × [0,1) under the lexicographic order, and the long line, with the order topology Definition
• The extended real line overlineℝ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined Definition
• The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua Definition
• The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ Definition
• The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A Definition
• Assuming the Axiom of Choice, compactness of [0,1] derived from the subbase lemma alone, using only the rays as a subbasis and the least upper bound property Example
• Infinite Ramsey for pairs gives a nondecreasing or nonincreasing subsequence of every real sequence Example
• ℝ ≈ P(ℕ) in ZF, by the Cantor set for one injection and by the cuts {q ∈ ℚ : q < x} for the other; so | ℝ | = 2^ℵ₀ under the Axiom of Choice Example
• ℝ with the half-open intervals [a,b) as a basis is not compact and, assuming the Axiom of Countable Choice, is Lindel"of, while its square is not Lindel"of, the antidiagonal being an uncountable closed discrete subspace Example
• The order topology on a totally ordered set, with the open rays as a subbasis, and its agreement with the usual topology of ℝ Example
• FALSE: every bounded sequence converges False statement
• FALSE: limits preserve strict inequalities False statement
• Every real sequence has a monotone subsequence (the peak / rising-sun lemma) Lemma
• Every subset of overlineℝ has a least upper bound and a greatest lower bound in overlineℝ, agreeing with the real supremum and infimum on nonempty sets bounded in ℝ Lemma
• For positive terms, null and divergence to +∞ are reciprocal Lemma
• Nonnegativity of a metric is a consequence of the other axioms, not an axiom Lemma
• The absolute value is compatible with limits Lemma
• The Cauchy-sequence reals are Archimedean Lemma
• The even and odd index maps and the alternating sequence: strictly increasing e, o with ℕ their disjoint union, and the unique (sₖ) with s₀ = 1, s_σ(k) = -sₖ, which satisfies |sₖ| = 1, s ∘ e ≡ 1 and s ∘ o ≡ -1 Lemma
• The rationals embed densely in the reals Lemma
• Conventions for sequences: indexing, eventually, lim, and rational ε Remark
• A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism Theorem
• The reals are complete Theorem
• The reals form a totally ordered field Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §5.4 (standard reference, not scraped)
• L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)