Compactness and Connectedness¶
Course Material
31/07 Lecture Slides
06/08 Lecture Slides
07/08 Lecture Slides
Introduction¶
A topological space \((X, \tau)\) is compact if for every \(\{V_i\}_{i \in I} \subseteq \tau\) such that
there is a finite subset \(\{i_1, \dots, i_n\} \subseteq I\) such that
A subset \(Y \subseteq X\) is compact if and only if it is compact in the subspace topology.
Covers and Compactness¶
Theorem: The interval \([0, 1]\) is compact.
Theorem: Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be compact topological spaces. Then \(X \times Y\) is compact (in the product topology).
Corollary: Sets of the form \([a, b]^n \subseteq \mathbb{R}^n\) are compact (with the standard topology).
Theorem (Heine-Borel): A subset \(X \subseteq \mathbb{R}^n\) is compact if and only if it is closed and bounded.
Theorem (Bolzano-Weierstrass): Every bounded sequence of real numbers has a convergent subsequence.
Lemma: Every sequence of real numbers has a monotone subsequence.
Corollary: Every bounded sequence in \(\mathbb{R}^n\) has a convergent subsequence.
Proof
Prove follows by entry-wise application of Bolzano-Weierstrass.
Sequentially Compactness¶
A topological space \((X, \tau)\) is called sequentially compact if every sequence in \(X\) has a convergent subsequence. Similarly, a subset \(Y \subseteq X\) is sequentially compact if it is sequentially compact in the subspace topology.
Theorem: Let \(X\) be a subset of \(\mathbb{R}^n\). The following are equivalent:
- \(X\) is compact
- \(X\) is sequentially compact
- \(X\) is closed and bounded
Proof
- \((1) \iff (3)\) by Heine Borel.
- \((3) \implies (2)\) by Bolzano-Weierstrass.
- \((2) \implies (3)\) ...
Uniform Continuity¶
Theorem: Let \((X, \tau_X)\) and \((X, \tau_Y)\) be topological spaces. If \(f: X \to Y\) is continuous and \(X\) is compact, then \(f(X) \subseteq Y\) is compact.
Corollary (Min-max Theorem): Let \(f: [a, b] \to \mathbb{R}\) be a continuous function. Then \(f\) attains maximum and minimum values.
Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. A function \(f: X \to Y\) is said to be uniformly continuous if \(\forall \epsilon > 0, \exists \delta(\epsilon)\) such that \(d_Y(f(x'), f(x)) < \epsilon\) whenever \(d_X(x, x') < \delta\).
Theorem: Let \((X, d)\) be a compact metric space, and let \(f: X \to \mathbb{R}\) be a continuous function. Then \(f\) is uniformly continuous.
Compactness in Metric Spaces¶
A metric space \((X, d)\) is said to be totally bounded if for every \(\epsilon > 0\) there is a finite set \(\{ x_1, \dots, x_n \} \subseteq X\) such that
Similarly, a subset of a metric space is totally bounded if it is totally bounded with respect to the subset metric.
Theorem: Let \((X, d)\) be a metric space. The following are equivalent:
- \(X\) is compact
- \(X\) is sequentially compact
- \(X\) is complete and totally bounded
The Arzela-Ascoli Theorem¶
Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. A subset \(S \subseteq C(X, Y)\) is said to be:
- Pointwise equicontinuous if
- Uniformly equicontinuous if
Theorem: let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces, with \(X\) compact. Then \(S \subseteq C(X, Y)\) is pointwise equicontinuous if and only if it is uniformly equicontinuous.
Theorem (Arzela-Ascoli): A bounded subset of \((C[0, 1], \lVert \cdot \rVert_\infty)\) is totally bounded if and only if it is equicontinuous.
(More generally, \([0, 1]\) can be replace by any compact metric space).
Corollary: A subset of \((C[0, 1], \lVert \cdot \rVert_\infty)\) is compact if and only if it is closed, bounded and equicontinous.
Corollary: A uniformly bounded and equicontinuous sequence of functions on a closed interval \([a, b]\) has a uniformly convergent subsequence.
The Weierstrass Approximation Theorem¶
Theorem (Weierstrass Approximation Theorem): Let \(f\) be a continuous function on a closed, bounded interval \([a ,b]\). For any \(\epsilon > 0\), there is a polynomial function \(p(x)\) such that
Consider a function \(f \in C[0, 1]\). The \(n^{th}\) Bernstein polynomial for \(f\) is defined as
Let \(X\) and \(Y\) be sets. A set \(S\) of functions between \(X\) and \(Y\) is said to separate points if for every pair of distinct points \(x, y \in X\), there is a function \(f \in S\) such that \(f(x) \neq f(y)\).
Theorem (Urysohn's Lemma): Let \(X\) be a compact Hausdorff space. Then \(C(X, \mathbb{R})\) separates points.
An algebra of functions is a vector space of functions with pointwise operations which is also closed under pointwise multiplication of functions. An algebra of functions is called unital if it contains the constant function 1.
Theorem (Stone-Weierstrass Theorem): Let \(X\) be a compact Hausdorff space, and let \(A \subseteq C(X, \mathbb{R})\) be a unital subalgebra. Then \(A\) is dense with respect to \(\lVert \cdot \rVert_\infty\) if and only if A separates points.
An algebra of complex-valued functions is called a *-algebra if it is closed under pointwise complex conjugation.
Theorem (Stone-Weierstrass Theorem - complex version): Let \(X\) be a compact Hausdorff space and let \(A \subseteq C(X, \mathbb{C})\) be a unital *-subalgebra. Then \(A\) is dense with respect to \(\lVert \cdot \rVert_\infty\) if and only if \(A\) separates points.
Theorem (Tychonoff's Theorem): Let \(\{(X_i, \tau_i)\}_{i \in I}\) be a collection of compact topological spaces. Then \(\prod_{i \in I} X_i\) is compact in the product topology.
Theorem: Let \(H\) be a Hilbert space. The closed unit ball in \(H\) (i.e. \(\{\mathbf{x} \in \mathbf{H} : \lVert \mathbf{x} \rVert \leq 1\}\)) is compact in the weak topology.