4.4 KiB
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 0d(x, y) = 0 \; \text{ if and only if } x = yd(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
d(x, y) = \begin{cases}
0 \; \text{ if } x = y \\
1 \; \text{ if } x \neq y
\end{cases}
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
V_\varepsilon(a) = {x \in S | d(a, x) < \varepsilon}
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},
n \geq K \Rightarrow d(x_n, x) \leq \varepsilon
Theorem. Let (x_n) be a sequence in metric space (S, d). Then,
(x_n)converges toxif and only if every $\varepsilon$-neighborhood ofxcontains all but finitely many terms of(x_n).- If
(x_n) \rightarrow xand(x_n) \rightarrow x', thenx = 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},
m, n \geq H \Rightarrow d(x_n, x_m) < \varepsilon
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.