Metric space Study Guide

📖 Core Concepts Metric space: an ordered pair \((M,d)\) where \(M\) is a set and \(d:M\times M\to\mathbb R\) satisfies Identity \(d(x,x)=0\) Positivity \(x\neq y\Rightarrow d(x,y)>0\) Symmetry \(d(x,y)=d(y,x)\) Triangle inequality \(d(x,z)\le d(x,y)+d(y,z)\). Open ball \(Br(x)=\{y\in M\mid d(x,y)<r\}\). Open set: a set that contains an open ball around each of its points (open balls form a basis). Convergence: \((xn)\to x\) iff \(\forall\varepsilon>0\,\exists N\) s.t. \(d(xn,x)<\varepsilon\) for \(n\ge N\). Cauchy sequence: \(\forall\varepsilon>0\,\exists N\) s.t. \(d(xm,xn)<\varepsilon\) for \(m,n\ge N\). Complete space: every Cauchy sequence converges to a point of the space. Compactness (metric version): Every open cover has a finite subcover ⇔ every sequence has a convergent subsequence ⇔ space is complete + totally bounded. Uniform continuity: \(\delta\) can be chosen independently of the point \(x\). Lipschitz map: \(d2(f(x),f(y))\le K\,d1(x,y)\) for some constant \(K\). Contraction: Lipschitz with \(0\le K<1\); Banach Fixed‑Point Theorem guarantees a unique fixed point in a complete space. 📌 Must Remember Metric axioms are necessary and sufficient for a distance function to be a metric. Discrete metric: \(d(x,y)=0\) iff \(x=y\), else \(1\). Standard metrics on \(\mathbb R^2\): Euclidean \(d2(x,y)=\sqrt{(x1-y1)^2+(x2-y2)^2}\) Manhattan \(d1(x,y)=|x1-y1|+|x2-y2|\) Chebyshev \(d\infty(x,y)=\max\{|x1-y1|,|x2-y2|\}\). Heine–Cantor: every continuous map from a compact metric space is uniformly continuous. Banach Fixed‑Point: contraction on a complete metric space → unique fixed point, iterates converge geometrically. Compact ⇔ complete + totally bounded (useful for proving compactness). Hausdorff distance \(dH(S,T)=\inf\{\varepsilon>0\mid S\subseteq T\varepsilon,\ T\subseteq S\varepsilon\}\) defines a metric on compact subsets. Gromov–Hausdorff distance: infimum of Hausdorff distances between isometric copies inside a common space. 🔄 Key Processes Checking a function is a metric Verify the four axioms for arbitrary \(x,y,z\). Proving a set is open Show each point \(x\) has an \(r>0\) with \(Br(x)\subseteq\) the set. Showing completeness Take an arbitrary Cauchy sequence, prove it converges using the definition of the space (e.g., monotone bounded sequence in \(\mathbb R\)). Establishing compactness Either: (a) extract a convergent subsequence from any sequence (sequential compactness), or (b) prove the space is complete and totally bounded. Applying Banach Fixed‑Point Verify the map is a contraction (\(K<1\)) and the domain is complete; then iterate \(x{n+1}=f(xn)\) to obtain the fixed point. Computing Hausdorff distance (for compact sets) Determine the smallest \(\varepsilon\) such that each point of one set lies within \(\varepsilon\) of the other set. 🔍 Key Comparisons Metric vs. Pseudometric – Metric: \(d(x,y)=0 \Rightarrow x=y\); Pseudometric: may have distinct points at distance 0. Metric vs. Quasimetric – Metric: symmetric; Quasimetric: may have \(d(x,y)\neq d(y,x)\). Uniform vs. Ordinary Continuity – Uniform: \(\delta\) depends only on \(\varepsilon\); ordinary: \(\delta\) may depend on the point. Contraction vs. Non‑expanding Lipschitz – Contraction: \(K<1\); Non‑expanding: \(K\le1\). Total boundedness vs. Boundedness – Bounded: all points fit in a single ball of radius \(R\); Total boundedness: can be covered by finitely many balls of any radius \(r>0\). ⚠️ Common Misunderstandings “All bounded spaces are totally bounded.” – False; e.g., infinite‑dimensional Banach spaces are bounded but not totally bounded. “If a sequence converges, it must be Cauchy.” – True, but the converse needs completeness of the space. “Uniform continuity implies Lipschitz continuity.” – Not necessarily; Lipschitz is stronger (requires a global linear bound). “Discrete metric makes every set open and closed.” – Correct, but it does not make the space compact unless the underlying set is finite. “Hausdorff distance is always finite.” – Only guaranteed for compact subsets; arbitrary subsets may yield infinite distance. 🧠 Mental Models / Intuition Metric axioms as a “road map”: Identity = staying at the same spot, Positivity = you can’t have negative distance, Symmetry = roads work both ways, Triangle inequality = the shortcut is never longer than the detour. Compactness = “no holes & no infinite escape routes.” Think of a rubber sheet you can stretch but never tear; every sequence must eventually settle down somewhere on the sheet. Total boundedness = “finite covering at any resolution.” Imagine zooming in: no matter how close you look, you can still cover the whole set with finitely many lenses of that size. Banach Fixed‑Point = “pulling a rubber band tighter and tighter.” Each iteration brings points strictly closer, guaranteeing they meet at a single point. 🚩 Exceptions & Edge Cases Incomplete spaces: \((0,1)\) with \(|x-y|\) lacks its endpoints; rational numbers \(\mathbb Q\) miss irrational limits. Extended metrics: allow distance \(\infty\); can be “compressed” to a real‑valued metric via \(\phi(t)=\frac{t}{1+t}\). Uncountable product of non‑trivial metric spaces: not metrizable (fails first‑countability). Quotient construction: yields only a pseudometric in general; becomes a metric only when distinct equivalence classes cannot have zero distance. 📍 When to Use Which Choose Euclidean vs. Manhattan vs. Chebyshev – depends on geometry of the problem: Euclidean for true geometric distance, Manhattan for grid‑like movement (city blocks), Chebyshev when only the largest coordinate difference matters (chess king moves). Uniform continuity vs. Lipschitz – use uniform continuity when only existence of a global \(\delta\) matters (e.g., on compact domains). Use Lipschitz when you need an explicit linear bound (e.g., error estimates). Banach Fixed‑Point – apply when you can show a map is a contraction on a complete space; otherwise look for Schauder or other fixed‑point theorems. Hausdorff distance – appropriate for comparing shapes, subsets, or graphs when both are compact. Product metric – select a norm \(N\) that matches the topology you need; Euclidean norm gives the standard product topology, any norm that is monotone in each coordinate works as well. 👀 Patterns to Recognize Metric‑space proofs often start with “Let \(\varepsilon>0\) … choose \(N\) …” – classic epsilon‑\(N\) structure for convergence, continuity, Cauchy. Compactness arguments: look for “finite subcover” or “sequential compactness” as interchangeable tools. Contraction verification: locate a constant \(K<1\) such that \(d(f(x),f(y))\le K\,d(x,y)\) for all \(x,y\). Total boundedness: whenever a problem mentions covering by balls of radius \(\varepsilon\), think “totally bounded”. Hausdorff distance calculations: often reduced to “max of two one‑sided distances”. 🗂️ Exam Traps Confusing bounded with totally bounded – an answer choice that says “every bounded metric space is compact” is false; missing the total boundedness requirement. Assuming every continuous map is uniformly continuous – only true on compact domains (Heine–Cantor). Mistaking a pseudometric for a metric – a statement that “\(d(x,y)=0\) implies \(x=y\)” is required for a true metric. Choosing the wrong product norm – any norm that is not monotone may produce a different topology; an answer claiming “any norm works for the product topology” is misleading unless monotonicity is mentioned. Banach Fixed‑Point misuse – applying it on a space that isn’t complete (e.g., \((0,1)\)) invalidates the theorem. Hausdorff distance on non‑compact sets – may be infinite; an answer that claims it’s always finite is incorrect.