REFERENCE SHEET

Metric compactness & fixed points

Compactness controls arbitrary covers; contraction controls successive approximations.

Before this Completeness & sequences, Limits & continuity

d(Tx,Ty) ≤ q d(x,y), 0≤q<1

Definitions

Metric

A nonnegative distance d with d(x,y)=0 iff x=y, symmetry, and the triangle inequality.

Compactness

Every open cover has a finite subcover. In metric spaces, this is equivalent to every sequence having a convergent subsequence with limit in the space.

Contraction

A self-map T satisfying the displayed bound with one fixed q<1 for all pairs.

Key theorems

Heine–Borel

A subset of ℝⁿ with its usual metric is compact exactly when it is closed and bounded.

Continuous images

The continuous image of a compact space is compact. A continuous real-valued function on a nonempty compact space attains extrema.

Banach fixed-point theorem

A contraction on a nonempty complete metric space has a unique fixed point. Starting anywhere, xₙ₊₁=T(xₙ) converges to it, with d(xₙ,x*)≤qⁿd(x₀,x₁)/(1−q).

Notable applications

Numerical equations

A contraction gives existence, uniqueness, an iteration, and an explicit error bound.

Uniform control

Compactness often turns local continuity statements into uniform estimates.

Work it out

Try before revealing

01 / Compute

For T(x)=(x+2)/3 on ℝ, find its fixed point and contraction constant.

Hint

Solve x=T(x) and compare Tx−Ty.

Solution

The fixed point is 1. Since |Tx−Ty|=|x−y|/3, q=1/3.

02 / Counterexample

Give a continuous function on (0,1) whose image is unbounded.