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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 \epsilon > 0, the $\epsilon$-neighborhood or $\epsilon$-ball of a point a \in S is the set
V_\epsilon(a) = {x \in S | d(a, x) < \epsilon}
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 \epsilon > 0 so that V_\epsilon(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 $\epsilon$-neighborhood of c contains some point a \in A such that a \neq c.
Theorem. Every $\epsilon$-neighborhood of a point is an open set.