Unit 4

Space and closeness

Distance, completeness, compactness: the bridge to analysis and topology.

5 chapters written · 32 definitions · 55 theorems · builds on Unit 1, Unit 2

The story

Unit 1 measured closeness with $|x - y|$ on the number line. Here distance becomes an idea of its own: anything that satisfies three rules is a metric, and the same proofs then work for points in the plane, for functions, and for strings of bits.

The unit builds open and closed sets (4.1), completeness and the fixed point theorem that powers existence proofs (4.2), compactness, the property that makes maxima exist (4.3), connectedness (4.4), and continuity described by open sets alone (4.5), which is where topology (MATH 551) begins.

Chapters

The theorem chain

Every theorem in the unit, in the order it is proved. Each one is used by the ones after it. The label says whether it is on the 375/376 exams (T), an Apostol extra (A), or beyond both (B).

4.1 Metric and normed spaces; open and closed sets

Theorem · B A norm gives a metric If $\lVert \cdot \rVert$ is a norm on $V$ , then $d(x, y) = \lVert x - y \rVert$ is a metric on $V$ .
Theorem · B The standard norms are norms $\lVert \cdot \rVert_1$ , $\lVert \cdot \rVert_2$ and $\lVert \cdot \rVert_\infty$ are norms on $\mathbb R^n$ , and the sup norm is a norm on $C[a, b]$ .
Theorem · B The discrete metric and the Hamming distance are metrics The discrete metric on any set, and $d_H$ on bit strings of length $n$ , satisfy (M1), (M2) and (M3).
Proposition · T Open balls are open Every open ball $B(x, r)$ is an open set.
Theorem · T Unions and finite intersections of open sets $\varnothing$ and $X$ are open; any union of open sets is open; the intersection of finitely many open sets is open.
Corollary · T Intersections and finite unions of closed sets $\varnothing$ and $X$ are closed; any intersection of closed sets is closed; the union of finitely many closed sets is closed.
Theorem · B Interior, closure and boundary $\operatorname{int} A$ is the largest open set inside $A$ ; $\overline A$ is the smallest closed set containing $A$ and equals $A$ together with its limit points; $\partial A = \overline A \setminus \operatorname{int} A$ .
Theorem · B A closed set contains its limit points A set $A$ is closed if and only if every limit point of $A$ belongs to $A$ .
Theorem · B Comparing the 1-, 2- and max-norms For every $x \in \mathbb R^n$ : $\lVert x \rVert_\infty \le \lVert x \rVert_2 \le \lVert x \rVert_1 \le \sqrt n\,\lVert x \rVert_2 \le n\,\lVert x \rVert_\infty$ .
Theorem · B Equivalent norms give the same open sets If two norms on $V$ are equivalent, a set is open for one iff it is open for the other, so they also have the same closed sets, interiors, closures and boundaries.

4.2 Convergence, completeness and the contraction mapping theorem

Theorem · B Limits in a metric space are unique A sequence in a metric space converges to at most one point.
Theorem · B Convergence in R^n is coordinatewise A sequence of vectors converges in $\mathbb R^n$ (for any of the 1-, 2- or max-distances) if and only if each coordinate converges as a sequence of real numbers.
Theorem · B Convergence in the sup metric is uniform convergence For continuous $f_n$ and $f$ on $[a, b]$ : $d_\infty(f_n, f) \to 0$ if and only if $f_n \to f$ uniformly on $[a, b]$ .
Theorem · B Closed sets and limits of sequences A point lies in the closure of $A$ if and only if some sequence in $A$ converges to it. Hence $A$ is closed if and only if the limit of every convergent sequence of points of $A$ lies in $A$ .
Theorem · B Convergent sequences are Cauchy If $x_n \to x$ in a metric space, then $(x_n)$ is Cauchy.
Theorem · B R^n is complete $\mathbb R^n$ with the 1-, 2- or max-distance is complete.
Theorem · B C[a, b] with the sup metric is complete Every Cauchy sequence of continuous functions on $[a, b]$ , in the sup metric, converges uniformly to a continuous function.
Theorem · B Closed subsets of complete spaces are complete Inside a complete space, a subset with the same distance is complete if and only if it is closed.
Theorem · B Derivative test for a contraction If $g$ is differentiable on an interval $I$ and $\lvert g'(x) \rvert \le q < 1$ for every $x \in I$ , then $\lvert g(x) - g(y) \rvert \le q\lvert x - y \rvert$ for all $x, y \in I$ .
Theorem · B Banach fixed point theorem A contraction $T$ of a non-empty complete metric space has exactly one fixed point $x^*$ ; from any start $x_0$ the iterates $x_{n+1} = T x_n$ converge to $x^*$ , with $d(x_n, x^*) \le \dfrac{q^n}{1 - q}\,d(x_1, x_0)$ .
Theorem · B Newton's method is a local contraction

4.3 Compactness

Theorem · B Sequentially compact sets are closed and bounded In any metric space, a sequentially compact set is closed and bounded.
Theorem · B Bolzano–Weierstrass in R^n Every closed bounded subset of $\mathbb R^n$ is sequentially compact: each sequence in it has a subsequence converging to a point of it.
Theorem · B Compact sets are closed and bounded In any metric space, every compact set is closed and bounded.
Lemma · B Closed subsets of compact sets are compact If $Q$ is compact and $F \subseteq Q$ is closed, then $F$ is compact.
Theorem · B Heine–Borel theorem A subset of $\mathbb R^n$ is compact if and only if it is closed and bounded.
Lemma · B Sequentially compact implies totally bounded A sequentially compact metric space can be covered by finitely many balls of any given radius.
Lemma · B Lebesgue number lemma
Theorem · B Compact equals sequentially compact In a metric space, a set is compact if and only if it is sequentially compact.
Theorem · B Compact equals complete and totally bounded A metric space is compact if and only if it is complete and totally bounded.
Proposition · B The unit ball of C[0,1] is closed and bounded but not compact In $C[0, 1]$ with the sup metric, the closed unit ball $\{f : \max|f| \le 1\}$ is closed and bounded and has a sequence with no convergent subsequence, so it is not compact.
Lemma · B Continuous maps send convergent sequences to convergent sequences If $f$ is continuous at $a$ and $x_n \to a$ , then $f(x_n) \to f(a)$ .
Theorem · B Continuous image of a compact set is compact If $K$ is compact and $f : K \to Y$ is continuous, then $f(K)$ is compact.
Theorem · B Extreme value theorem on a compact set A continuous real function on a non-empty compact set attains a maximum and a minimum.
Theorem · B Continuous on a compact set implies uniformly continuous A continuous map from a compact metric space to any metric space is uniformly continuous: one $\delta$ works for the whole set.

4.4 Connectedness and path-connectedness

Lemma · B Open sets inside a subset $U$ is open in $A$ if and only if $U = A \cap W$ for some open set $W$ of $X$ .
Proposition · B Connected means no proper clopen subset A space $A$ is connected if and only if the only subsets that are both open and closed in $A$ are $\varnothing$ and $A$ .
Lemma · B A connected subset lies on one side of a separation If $Z = P \cup Q$ is a separation and $C \subseteq Z$ is connected, then $C \subseteq P$ or $C \subseteq Q$ .
Theorem · B Connected subsets of the line are the intervals A subset of $\mathbb R$ is connected if and only if it is an interval.
Lemma · B Continuous maps pull open sets back to open sets If $f : X \to Y$ is continuous and $W$ is open in $Y$ , then $f^{-1}(W) = \{x : f(x) \in W\}$ is open in $X$ .
Theorem · B The continuous image of a connected set is connected If $X$ is connected and $f : X \to Y$ is continuous, then $f(X)$ is connected.
Corollary · B Intermediate value theorem from connectedness If $X$ is a connected metric space and $f : X \to \mathbb R$ is continuous, then $f$ takes every value between any two of its values.
Theorem · B Path-connected implies connected A path-connected space is connected.
Theorem · B Open connected sets in R^n are path-connected If $U \subseteq \mathbb R^n$ is open and connected, then any two points of $U$ can be joined by a path in $U$ .
Lemma · B Closure of a connected set is connected If $A$ is connected and $A \subseteq B \subseteq \overline A$ , then $B$ is connected.
Proposition · B The topologist's sine curve $S = G \cup L$ is connected, but there is no path in $S$ from $(0, 0)$ to $(1, \sin 1)$ .
Theorem · B Zero gradient on a connected open set means constant If $U \subseteq \mathbb R^n$ is open and connected and $f$ is differentiable on $U$ with $\nabla f = O$ everywhere, then $f$ is constant on $U$ .

4.5 Continuity between metric spaces

Theorem · B Three faces of continuity For $f : X \to Y$ between metric spaces: $f$ is continuous (with $\varepsilon$ and $\delta$ ) iff $x_n \to a$ implies $f(x_n) \to f(a)$ iff the preimage of every open set is open iff the preimage of every closed set is closed.
Theorem · B Building continuous maps Compositions of continuous maps are continuous; sums, multiples, products and quotients (with non-zero denominator) of continuous real functions are continuous; a map into $\mathbb R^m$ is continuous iff each of its $m$ coordinate functions is; coordinate projections are continuous.
Theorem · B Lipschitz implies uniformly continuous A Lipschitz map is uniformly continuous, hence continuous.
Theorem · B Uniform continuity between metric spaces
Theorem · B Properties preserved by homeomorphisms A homeomorphism sends open sets to open sets, convergent sequences to convergent sequences, compact sets to compact sets and connected sets to connected sets, in both directions.
Proposition · B A continuous bijection from a compact space is a homeomorphism If $K$ is compact and $f : K \to Y$ is a continuous bijection onto a metric space, then $f^{-1}$ is continuous, so $f$ is a homeomorphism.
Proposition · B Equivalent norms give the same continuous maps If two norms on $V$ are equivalent, the identity map between the two normed spaces is a bi-Lipschitz homeomorphism, so a map into or out of $V$ is continuous for one norm iff it is for the other.
Theorem · B All norms on R^n are equivalent For any norm $\lVert \cdot \rVert$ on $\mathbb R^n$ there are constants $0 < m \le M$ with $m\lVert x \rVert_1 \le \lVert x \rVert \le M\lVert x \rVert_1$ for all $x$ .

Terms this unit defines

Unit review

A mixed set of 12 quick checks drawn at random from the 45 in this unit's chapters. Each one says which chapter it comes from, so a miss tells you what to reread. Press "New set" for another mix.

Where it leads

Units that build on this one: Unit 5, Differential calculus in many variables, Unit 7, Differential equations, Unit 13, Geometry and topology.