Alphabeta Math

Completeness, Completion, and Uniform Continuity: Examples and Counterexamples

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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The completion of ℚ under the usual metric is ℝ

Example

The map x ↦ (x + 2/x)/2 is a contraction of [1,2] with fixed point √2, and the a priori bound gives the error after n steps

Example

A Lipschitz function on ℚ extends uniquely to a Lipschitz function on ℝ with the same constant

Example

√· on [0,∞) is uniformly continuous and exactly 1/2-Hölder, and is not Lipschitz

Example

The bounded real-valued functions on a set, with the supremum metric, form a complete metric space

Counterexample

x ↦ x/2 maps (0,1] into itself, is a 1/2-contraction, and has no fixed point

Counterexample

x ↦ x + 1/x on [1,∞) strictly decreases every distance and has no fixed point

Counterexample

On ℕ with d(m,n) = 1 + 1/(m+n) for m ≠ n the sets {n, n+1, …} are nested, closed, bounded and complete with empty intersection

Counterexample

On the positive integers the metrics |m-n| and |1/m - 1/n| both induce the discrete topology, and only the first is complete

Counterexample

On (0,∞) the metrics |x-y| and |1/x - 1/y| share their topology and not their Cauchy sequences

Counterexample

x ↦ 1/x is continuous on (0,1) and sends the Cauchy sequence (1/(k+2))_k ≥ 0 to an unbounded one

Counterexample

x ↦ √x is a uniformly continuous bijection of [0,∞) onto itself whose inverse x ↦ x² is not uniformly continuous

Sources