Connectedness and Compactness

Connectedness and Compactness: “Unbreakability” and “Finiteness” in Topology

Connectedness: it is impossible to “break” the space

A connected space cannot be split into two non-empty, disjoint open sets. Formally: X is connected if X = U ∪ V, U ∩ V = ∅, U, V open ⇒ U = ∅ or V = ∅.

Intuition: a connected space is “all in one piece.” A disconnected space is “broken up” into parts.

Examples: ℝ is connected; removing one point makes it disconnected: ℝ {0} = (−∞, 0) ∪ (0, +∞). The union (0,1) ∪ (2,3) is disconnected (an explicit division into two open sets). The irrational numbers ℚ^c are disconnected as a subspace of ℝ. GL(2,ℝ) is disconnected: matrices with det > 0 and det < 0 form two different components; SL(2,ℝ) is connected.

Component of connectedness: The maximal connected subset. For disconnected X: X = ⊔_α C_α. The number of components β₀ is a topological invariant.

Path connectedness: Any two points can be joined by a continuous path. Path connectedness ⇒ connectedness. The converse is false: the “topological sine curve” {(x, sin(1/x)) : x > 0} ∪ {0} × [−1,1] is connected but not path connected.

Intermediate Value Theorem: A consequence of connectedness of ℝ. If f: [a,b] → ℝ is continuous and f(a) < 0 < f(b), then ∃c ∈ (a,b): f(c) = 0. Proof: the image of the connected [a,b] is connected in ℝ, that is, an interval → contains 0.

Compactness: “finiteness” without finiteness

A compact space X: from any open cover X = ∪_α U_α one can select a finite subcover X = U₁ ∪ ... ∪ Uₙ.

Intuition: a compact space is “controlled” by a finite number of “pieces.” In metric spaces, this is equivalent to boundedness and closedness (the Heine–Borel theorem for ℝⁿ).

Heine–Borel Theorem: A subset of ℝⁿ is compact ↔ it is closed and bounded.

Sequential compactness: From any sequence, one can select a convergent subsequence. In metric spaces, equivalent to compactness.

Tychonoff’s Theorem: The product of compact spaces is compact (in any product, with the axiom of choice). A very important result, used in functional analysis.

Consequences of compactness

The continuous image of a compact is compact: If f: X → Y is continuous and X is compact ⇒ f(X) is compact. Consequence: a continuous function on a compact is bounded.

Weierstrass Extreme Value Theorem: A continuous function on a compact space achieves min and max. Generalization of the min/max theorem on a closed bounded interval.

Examples of application: The sphere S² is compact ⇒ any continuous function S² → ℝ is bounded and attains extrema (useful in physics: potential on a closed surface). Phase space (compact) is used in statistical mechanics to correctly define the microcanonical ensemble.

Local compactness and one-point compactification

Local compactness: Every point has a compact closed neighborhood. Examples: ℝⁿ (balls of finite radius are compact), smooth manifolds.

Alexandroff one-point compactification: For locally compact Hausdorff X: X* = X ∪ {∞}. The open sets in X* are the open sets in X plus complements of compacts in X. X* is compact Hausdorff.

Examples: ℝ* = S¹ (the real line “closed up” into a circle—stereographic projection!). ℝⁿ* = Sⁿ. ℂ* = S² (Riemann projective plane / Riemann sphere).

Numerical example: One-point compactification of ℕ = {1, 2, 3, ...}: ℕ* = ℕ ∪ {∞}. An open neighborhood of ∞ = {∞} ∪ {n > N} for any N. The sequence 1, 2, 3, ... converges to ∞ in ℕ*.

Simple connectedness (fundamental group)

X is simply connected if π₁(X) = 0—every loop contracts to a point. Examples: ℝⁿ (any loop can be shrunk to a point), Sⁿ (n ≥ 2), disk D².

Not simply connected: S¹ (π₁ = ℤ—a loop around—cannot be contracted), torus T² (π₁ = ℤ²).

Application: Classification of electrical circuits: a circuit without “loops” (π₁ = 0) has a unique potential. “Loops” create the possibility of nontrivial cyclic currents (Kirchhoff’s law).

Connectedness and compactness in analysis and variational calculus

Compactness and connectedness are the workhorses of modern analysis. The Arzelà–Ascoli theorem on the compactness of a family of equicontinuous functions on a compact lies at the foundation of proofs of existence of solutions of differential equations—via the limit passage in approximating sequences. In variational calculus, existence of the minimum of a functional (e.g., length of a curve or area of a surface) is guaranteed by the compactness of the admissible class of functions in a suitable topology: the weak topology in Hilbert space makes closed convex sets compact, which ensures existence of a minimum even for infinite-dimensional problems. The direct method of variational calculus of Hilbert–Lebesgue relies precisely on sequential compactness for extracting a convergent subsequence of minimizing functions. In topology of finite groups, compactness and connectedness of a Lie group determine the structure of its representations: compact groups (SO(n), U(n), SU(n)) have only finite-dimensional unitary irreducible representations, whereas noncompact (GL(n,ℝ), SL(2,ℝ)) allow infinite-dimensional.

Compactness and completeness in computational methods

Compactness and completeness are key properties for justifying numerical methods. Tychonoff’s theorem (the product of compact spaces is compact) is the basis for the existence of solutions of variational problems in functional spaces: existence of a minimum of a functional over a compact set guarantees convergence of finite element methods. In optimal control theory, Filippov’s theorem on existence of optimal control uses compactness of the admissible set of states and controls. In machine learning, compactness of the parameter space (under L2-regularization) guarantees that the sequence of steps of gradient descent has convergent subsequences—this substantiates the applicability of iterative algorithms. In data compression, Vitali’s theorem (the integral over a compact set can be approximated by a step function) underlies the JPEG and MPEG algorithms. Completeness of Sobolev spaces H^m(Ω) is the foundation of the theory of elliptic partial differential equations: functional spaces in which weak solutions are sought must be complete Banach spaces, otherwise the finite element method does not converge to the correct answer.

Assignment: (a) Prove that the continuous image of a connected set is not necessarily compact. But the image of a compact set is compact. Prove the latter. (b) Brouwer’s fixed point theorem: any continuous f: Dⁿ → Dⁿ has a fixed point. Prove for D¹ = [0,1] (via the intermediate value theorem). (c) One-point compactification of ℂ: ℂ* ≅ S². Explicitly describe the stereographic projection φ: S² {N} → ℂ.