Metric Spaces: Complete Metric Spaces

Cauchy criterion: for all \(\epsilon > 0\) there exists \(N_\epsilon\) such that \(\rho(x_n, x_{n^\prime}) < \epsilon\) for all \(n, n^\prime > N_\epsilon\). A sequence satisfying this criterion is called a Cauchy sequence or fundamental.

Theorem 1

Every convergent sequence \(\{x_n\}\) is fundamental.

Complete if every Cauchy sequence in \(R\) converges to an element of \(R\). Incomplete otherwise.

Theorem 2 (Nested sphere theorem) Complete if and only if every nested sequence \[\{S_n\} = \{S_\infty [x_n, r_n]\}\] of closed spheres such that \(r_n \rightarrow 0\) as \(n \rightarrow \infty\) has a nonempty intersection \(\cap_{n=1}^\infty S_n\).

Theorem 3 (Baire) A complete metric space cannot be represented as the union of a countable number of nowhere dense sets.

\(R^*\) completion of \(R\) if \(R \subseteq R^*\) and \([R] = R^*\).

Theorem 4 Every metric space \(R\) has a completion. This completion is unique to within an isometric mapping carrying every point \(x \in R\) into itself.

Problems

  1. Prove that the limit \(f(t)\) of a uniformly convergent sequence of functions \(\{f_n(t)\}\) continuous on \([a, b]\) is itself a function continuous on \([a, b]\).

    Hint. Clearly \[|f(t) - f(t_0)| \leq |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)|\] where \(t, t_0 \in [a, b]\). Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large \(n\). Then use the continuity of \(f_n(t)\) to make the second term small for \(t\) sufficiently close to \(t_0\).

    Do as the hint says.

  2. Prove that the space \(m\) is complete.

    \(\rho(x_n, x_{n^\prime}) = \sup_k (x_{n_k} - x_{{n^\prime}_k}) < \epsilon\)

  3. Prove that if \(R\) is complete, then the intersection \(\cap_{n = 1}^\infty S_n\) from Theorem 2 consists of a single point.

    \(R\) complete, so limit in \(R\) by definition. And by Theorem 2, intersection nonempty. Need to prove uniqueness. Assume for contradiction there exists a second distinct point, such that the distance between the two points is nonzero. Then one of the spheres will not contain the second point, as the sphere will be smaller than the distance. As the second point is arbitrary, we have our result.

  4. The diameter of a subset \(A\) of a metric space \(R\) is \[d(A) = \sup_{x, y \in A} \rho(x, y)\] Suppose \(R\) complete, and let \(\{A_n\}\) be a sequence of nested of closed subsets of \(R\). Suppose further that \(\lim_{d \rightarrow \infty} d(A_n) = 0\). Prove that the intersection \(\cap_{n = 1}^\infty A_n\) is nonempty.

    Same as the nested sphere theorem. Also use definition of complete.

  5. A subset \(A\) of a metric space \(R\) is said to be bounded if its diameter \(d(A)\) is finite. Prove that the union of a finite number of bounded sets is bounded.

    There’ll always still be a lowest and hightest number.

  6. Give an example of a complete metric space \(R\) and a nested sequence \(\{A_n\}\) of closed subsets of \(R\) such that \(\cap_{n=1}^\infty A_n = \emptyset\). Reconcile this example with Problem 4.

    Take all of the subsets to be the empty set. The diameter limit no longer holds.

  7. Prove that a subspace of a complete metric space \(R\) is complete if and only if it is closed.

    Both ways by definitions.

  8. Prove that the real line equipped with the distance \[\rho(x, y) = |\arctan x - \arctan y|\] is an incomplete metric space.

    \(\frac{\pi}{2}\).

  9. Give an example of a complete metric space homeomorphic to an incomplete metric space.

    \(f(x) = \frac{2}{\pi} \arctan x\)

  10. Construct the real number system.

    Hint. If \(\{x_n\}\) and \(\{y_n\}\) are Cauchy sequences of rational numbers serving as ‘representatives’ of real numbers \(x^*\) and \(y^*\), respectively, define \(x^* + y^*\) as the real number with representative \(\{x_n + y_n\}\).

    Also need to define the products, and verify that the usual axioms of arithmetic are satisfied.

Christian Christiansen