Metric Spaces¶
Job: quantitative topology — completeness, compactness criteria, uniform structure, and Baire category as used in analysis.
Core¶
Definition. A metric \(d:X\times X\to[0,\infty)\) satisfies \(d(x,y)=0\Leftrightarrow x=y\), symmetry, triangle inequality. Topology: open balls \(B(x,r)\) form a basis. Isometry preserves \(d\); uniformly continuous: \(\forall\varepsilon\exists\delta\) independent of basepoint.
| Concept | Definition / fact |
|---|---|
| Cauchy sequence | \(d(x_n,x_m)\to 0\) as \(n,m\to\infty\) |
| Complete | every Cauchy sequence converges in \(X\) |
| Completion \(\widehat{X}\) | unique complete space with \(X\) dense isometrically embedded |
| Totally bounded | \(\forall\varepsilon\): finite \(\varepsilon\)-net |
| Compact metric | complete + totally bounded \(\Leftrightarrow\) sequentially compact |
| Heine–Borel (\(\mathbb{R}^n\)) | closed bounded \(\Leftrightarrow\) compact |
| Lipschitz / Hölder | \(d(fx,fy)\le K\,d(x,y)^\alpha\) |
| Equivalent metrics | same topology; strongly equivalent: \(c\,d\le d'\le C\,d\) |
Completeness¶
Closed subset of complete \(\Rightarrow\) complete. Product of countably many complete metric spaces is completely metrizable (e.g. \(\mathbb{R}^{\mathbb{N}}\)).
Theorem (Baire category). \(X\) complete metric (or locally compact Hausdorff). Countable intersection of dense open sets is dense. Equiv.: \(X\) is not a countable union of nowhere dense sets.
Consequences: uniform boundedness (BANACH–Steinhaus), open mapping, existence of continuous nowhere differentiable functions (comeager in \(C[0,1]\)), residual sets in dynamical systems.
Compactness in metric spaces¶
Theorem. For metric \(X\), TFAE: (i) compact; (ii) sequentially compact; (iii) complete and totally bounded; (iv) every infinite set has an accumulation point.
Lebesgue number: for open cover of compact metric, \(\exists\delta\) s.t. any set of diam \(<\delta\) lies in some cover element — used in refining covers for \(\pi_1\) and homology.
Continuous functions¶
\(C(X,Y)\) with sup metric when \(Y\) bounded metric; if \(X\) compact, \(C(X,\mathbb{R})\) Banach under \(\|\cdot\|_\infty\).
Theorem (Arzelà–Ascoli). \(X\) compact metric, \(\mathcal{F}\subset C(X,\mathbb{R})\): relatively compact iff pointwise bounded and equicontinuous.
Theorem (Stone–Weierstrass). Subalgebra of \(C(X,\mathbb{R})\) (\(X\) compact Hausdorff) separating points, containing constants, dense in \(\|\cdot\|_\infty\). Complex version: closed under conjugation.
Proper metric spaces¶
Closed balls compact \(\Leftrightarrow\) proper (as a map to a point in coarse geometry language: inverse images of compacts are compact). Riemannian manifolds with geodesic distance: Hopf–Rinow.
See it¶
\(C[0,1]\) vs \(L^p\). \(C[0,1]\) complete in \(\|\cdot\|_\infty\), not in \(\|\cdot\|_1\); completion of \(C_c\) in \(L^p\) norm is \(L^p\). Metric completeness is the analytic content of Banach space hypotheses.
p-adic metric. \(d_p(x,y)=p^{-v_p(x-y)}\) on \(\mathbb{Q}\); completion \(\mathbb{Q}_p\). Ultrametric inequality \(d(x,z)\le\max(d(x,y),d(y,z))\) — balls are clopen; totally disconnected locally compact field.
Check¶
Why is \((0,1)\) not complete in the standard metric, and what is its completion?
Cauchy sequence \(x_n=1/n\) has no limit in \((0,1)\). Completion is \([0,1]\) (or isometrically, the abstract completion is homeomorphic to \([0,1]\)).
Show that infinite-dimensional normed spaces have noncompact closed unit ball (Riesz lemma sketch).
Riesz: for closed proper subspace \(Y\), \(\exists x\), \(\|x\|=1\), \(d(x,Y)\ge 1-\varepsilon\). Iterate to get sequence with \(d(x_n,x_m)\ge 1/2\); no convergent subsequence.
Pitfalls¶
- Topologically equivalent metrics need not share Cauchy sequences (e.g. \(d\) and \(\arctan d\) on \(\mathbb{R}\): same topology, different completeness — \(\mathbb{R}\) complete in \(d\), bounded metric not complete? Actually \((\mathbb{R},\arctan|x-y|)\) is not complete). Completeness is metric, not topological.
- Total boundedness \(\neq\) boundedness in infinite dimensions.
- Uniform continuity on noncompact sets fails (\(x\mapsto x^2\) on \(\mathbb{R}\)).
- Length spaces / geodesic structure is extra geometric data beyond the metric topology.
Refs¶
- Rudin, Principles of Mathematical Analysis
- Munkres, Topology
- Burago–Burago–Ivanov, A Course in Metric Geometry (length spaces)
Next¶
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