Chapter 11 - Metric Spaces
Section 11.4 - Netric Spaces
Definition. A metric on set \(S\) is a function \(d: S \otimes S \rightarrow \mathbb{R}\) that satifies the following properties for all \(x, y, z \in S\),
- \(d(x, y) \geq 0\)
- \(d(x, y) = 0 \; \text{ if and only if } x = y\)
- \(d(x, y) = d(y, x)\)
- \(d(x, y) \leq d(x, z) + d(z, y)\)
Definition. A metric space \((S, d)\) is a set \(S\), with elements called points, together with a metric \(d\).
Definition. With metric space \((S, d)\), if \(A \subset S\), then \((A, d)\) is a subspace of \((S, d)\).
Definition. The discrete metric is provided by
Definition. Let \((S, d)\) be a metric space. Then, for each \(\varepsilon > 0\), the \(\varepsilon\)-neighborhood or \(\varepsilon\)-ball of a point \(a \in S\) is the set
Definition. Let \((S, d)\) be a metric space. Then, a subset \(G \subseteq S\) is open if for each \(x \in G\), there exists some \(\varepsilon > 0\) so that \(V_\varepsilon(x) \subseteq G\).
Definition. Let \((S, d)\) be a metric space. Then, a subset \(G \subseteq S\) is closed if its complement \(C(G) = S - G = S \ F\) is closed.
Definition. Let \((S, d)\) be a metric space. A point \(c \in S\) is a *cluster point$ of a set \(A \subseteq S\) if every \(\varepsilon\)-neighborhood of \(c\) contains some point \(a \in A\) such that \(a \neq c\).
Theorem. Every \(\varepsilon\)-neighborhood of a point is an open set.
Theorem. The union of an arbitrary collection of open sets is open.
Theorem. The intersection of a finite collection of open sets is open.
Theorem. The union of finitely many closed sets is closed.
Theorem. The intersection of infinitely many closed sets is closed.
Theorem. A subset of a metric space is closed if and only if it contains all of its cluster points.
Definition. A sequence \((x_n)\) in a metric space \((S, d)\) converges to a point \(x \in S\) if given any \(\varepsilon > 0\), there exists a \(K \in \mathbb{N}\) such that given \(n \in \mathbb{N}\),
Theorem. Let \((x_n)\) be a sequence in metric space \((S, d)\). Then,
- \((x_n)\) converges to \(x\) if and only if every \(\varepsilon\)-neighborhood of \(x\) contains all but finitely many terms of \((x_n)\).
- If \((x_n) \rightarrow x\) and \((x_n) \rightarrow x'\), then \(x = x'\).
- If \((x_n)\) converges, then \((x_n)\) is bound.
Definition. A sequence \((x_n)\) in metric space \((S, d)\) is a Cauchy sequence if for every \(\varepsilon > 0\), there exists some \(H \in \mathbb{N}\) such that for any \(m, n \in \mathbb{N}\),
Definition. A metric space in which every Cauchy sequence converges is said to be complete.
Remark. \(\mathbb{R}\) is complete, but \(\mathbb{Q}\) is not.
Definition. Let \(A\) be a subset of metric space \((S, d)\). Then, an open cover of \(A\) is some collection of subsets \(\mathcal{G} = \{G_\alpha\}_{\alpha \in I}\), such that \(G_\alpha \subseteq S\) and \(A \subseteq \cup_{\alpha \in I} G_\alpha\). That is, \(A\) is contained within the union of all open subsets in \(\mathcal{G}\).
Definition. If \(\mathcal{G}' \subseteq \mathcal{G}\) is an open cover of \(A\), then \(\mathcal{G}'\) is a subcover of \(\mathcal{G}\).
Definition. Given \(K\) is a subset of metric space \((S, d)\), K is compact if every cover of \(K\) contains a finite subcover.
Theorem. If \(K\) is a compact subset of a metric space, then \(K\) is closed and bounded.
Theorem. Heine-Borel Theorem. In \(\mathbb{R}\), the convese is true. That is, if \(K \subseteq \mathbb{R}\) is closed and bounded, then it is compact.
Theorem. If \(K\) is a compact subset of a metric space, then every infinite subset of \(K\) has a cluster point.
Corollary. Bolzano-Weirstrass Theorem. Every bounded infinite subset of \(\mathbb{R}\) has a cluster point in \(\mathbb{R}\).
Definition. Let \((S, d)\) be a metric space, and \(A_1, A_2 \in S\) be subsets. Then, \(A_1, A_2\) are said to be separated if there exist disjoint open subsets \(U_1, U_2\) such that \(A_1 \subseteq U_1\) and \(A_2 \subseteq U_2\).
Definition. A subset \(C\) of metric space \((S, d)\) is said to be connected if it is not the union of nonempty separated subsets.
Theorem. A subset of \(\mathbb{R}\) is connected if and only if it is an interval.