Metric Spaces

Definition and Examples

Definition 1. We say $(X, d)$ is metric space if $X$ is a non-empty set, $d \colon X \times X \to [0, \infty)$ is such that

  1. $d(x, y) = 0$ if and only if $x = y$,
  2. $d(x, y) = d(y, x)$
  3. For every $x, y, z \in X$, $d(x, z) \leq d(x, y) + d(y, z)$

Remark 2. Elements of $X$ are called points, the function $d$ is called a distance function and $d(x, y)$ is called the distance between $x$ and $y$.

Example 3. $X = \R^d$ with $d(x, y) = \abs{x - y}$ is a metric space.

Example 4. $X = \R^d$, with $d(x, y) = \max_i \abs{x_i - y_i}$ is a metric space. (We will see later that this metric is equivalent to the standard Euclidean metric.)

Remark 5. Any subset $Y \subseteq X$ is a metric space (with the same distance function).

Example 6 (trivial metric). Let $X$ be any set and $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$. Then $(X, d)$ is a metric space.

Question 7. Let $X$ be the set of all Riemann integrable functions on $[0, 1]$, and define \begin{equation} d(f, g) = \int_0^1 \abs{f - g} \, dx \,. \end{equation} Is $(X, d)$ a metric space?

The Metric Topology

Fix a metric space $(X, d)$.

Definition 8. $B(x, r) = \set{ y \in X \st d(x, y) < r}$ is called the open ball centered at $x$ with radius $r$.

Definition 9. $B(x, r) = \set{ y \in X \st d(x, y) \leq r}$ is called the closed ball centered at $x$ with radius $r$.

Definition 10. We say $E \subseteq X$ is open if for every $x \in E$ there exists $r > 0$ such that $B(x, r) \subseteq E$.

Definition 11. We say $E \subseteq X$ is closed if $E^c$ is open. (Equivalently, for every $x \not\in E$, there exists $r > 0$ such that $B(x, r) \cap E = \emptyset$.)

Example 12. Open balls are open sets, closed balls are closed sets.

Question 13. Is any finite set closed?

Proposition 14.

  1. $\emptyset, X$ are both open and closed.
  2. An arbitrary union of open sets is open.
  3. The finite intersection of open sets is open.

Remark 15. A collection of sets satisfying the above three properties is said to form a topology on $X$.

Remark 16. By De Morgan’s law, a finite union of closed sets is closed, and an arbitrary intersection of closed is closed.

Definition 17. We say $p$ is a limit point of $E$ if every neighborhood of $p$ contains a point in $E$ that is not $p$.

Remark 18. A neighborhood of a point $p$ is any open set containing $p$.

Proposition 19. If $U \ni p$ is open and $p$ is a limit point of $E$, then $U \cap E$ has infinitely many points.

Proposition 20. A set $C$ is closed if and only if it contains all its limit points.

Question 21. Let $(X, d)$ be a metric space, and $Y \subseteq X$ be a non-empty subset. We know $(Y, d)$ is also a metric space.

  1. If $U \subseteq X$ is an open subset of $X$, must $U \cap Y$ be an open subset of $Y$?
  2. If $C \subseteq X$ is a closed subset of $X$, must $C \cap Y$ be a closed subset of $Y$?
  3. Are the answers to the first two parts different if we assume $Y$ is an open subset of $X$?
  4. Are the answers to the first two parts different if we assume $Y$ is an closed subset of $X$?

Closures and Interiors

Definition 22. If $E \subseteq X$, define the interior of $E$, denoted by $\mathring E$, by \begin{equation} \mathring E = \bigcup_{U \text{ open},~ U \subseteq E} U \,. \end{equation}

Definition 23. If $E \subseteq X$, define the closure of $E$, denoted by $\bar E$, by \begin{equation} \bar E = \bigcap_{C \text{ closed},~ C \supseteq E} C \,. \end{equation}

Definition 24. If $E \subseteq X$, define the boundary of $E$, denoted by $\partial E$, by $\partial E = \bar E - \mathring E$.

Proposition 25. $\mathring E = \set{x \in E \st \exists r > 0 \,\colon\, B(x, r) \subseteq E }$.

Proposition 26. $\partial E$ is the set of all limit points of $E$, and $\bar E = E \cup \partial E$.