Definition. A metric on a set X is a function: \rho: X\times X\to[0, \infty) such that

1) \rho(x, y)=0 iff x=y;

2) \rho(x, y)=\rho(y, x) for all x, y\in X;

3) \rho(x, x)\le\rho(x, y)+\rho(y, z) for all x, y, z\in X.

A set equipped with a metric is called a metric space.

 

Examples. 1)  The Euclidean distance \rho(x, y)=|x-y| is a metric on \mathbb R^n.

2) \rho_1(x, y)=\int_0^1|f(x)-g(x)|dx and \rho_\infty(f, g)=\sup_{0\le x\le 1}|f(x)-g(x)| are metrics on the space of continuous functions on [0, 1].

3) If \rho is a metric on X and A\subset X, then \rho|(A\times A) is a metric on A.

4) If (X_1, \rho_1) and (X_2, \rho_2) are metric spaces, the product metric \rho on X_1\times X_2 is given by

\rho((x_1, x_2),(y_1, y_2))=\max(\rho_1(x_1, y_1), \rho_2(x_2, y_2)).

5) Some other metrics used on X_1\times X_2:

\rho_1(x_1, y_1)+\rho_2(x_2, y_2), [\rho_1(x_1, y_1)^2+\rho_2(x_2, y_2)^2]^{1/2}.

Definition. Let (X, \rho) be a metric space. If x\in X and r>0, the (open) ball of radius r about x is B(r, x)=\{y\in X: \rho(x, y)<r\}.

A set E\subset X is open if for every x\in E, \exists r>0 such that B(r, x)\subset E, and closed if the complement is open.

Remark. X and \emptyset are both open and closed. The union of any family of open sets is open, and the intersection of any family of closed sets is closed. The intersection (resp. union) of any finite family of open (resp. closed) sets is open (resp. closed). Indeed, if U_1, U_2, ..., U_n are open and x\in\cap_1^n U_j, for each j, $\exists r_j>0$ such that B(r_j, x)\subset U_j, and then B(r, x)\subset\cap_1^n U_j where r=\min(r_1, ..., r_n), so \cap_1^n U_j is open.

Definition. If E\subset X, then the interior of E, denoted by E the union of all open sets U\subset E is the largest open set contained in E, which is called the interior of E and is denoted by E^0. The intersection of all closed sets F\supset E is the smallest closed set containing E; it is called the closure of E and is denoted by \bar{E}. E is said to be dense in X if \bar{E}=X, and nowhere dense if \bar{E} has empty interior. X is called separable if it has a countable dense subset. (For example, \mathbb Q^n is a countable dense subset of \mathbb R^n.) A sequence \{x_n\} in X converges to x\in X (symbolically: x_n\to x or \lim x_n=x) if \lim_{n\to\infty}\rho(x_n, x)=0.

Proposition. If X is a metric space, E\subset X, and x\in X, the following are equivalent:

1) x\in\bar{E}.

2) B(r, x)\cap E\not=\emptyset for all r>0.

3) There is a sequence \{x_n\} in E that converges to x.

Proof. If B(r, x)\cap E=\emptyset, then B(r, x)^c is a closed set containing E but not x, so x\not\in\bar{E}. Conversely, if $x\not\in\bar{E}$, since (\bar{E})^c is open $\exists r>0$ such that B(r, x)\subset(\bar E)^c\subset E^c. Thus 1) is equivalent to 2). If 2) holds, for each n\in\mathbb N, $\exists x_n\in B(n^{-1}, x)\cap E$, so that x_n\to x. On the other hand, if B(r, x)\cap E=\emptyset, then \rho(y, x)\ge r for all y\in E, so no sequence of E can converge to x. Thus 2) is equivalent to 3). \Box

Remark. If (X_1, \rho_1) and (X_2, \rho_2) are metric spaces, a map f: X_1\to X_2 is called continuous at x\in X_1 if for every \epsilon>0, \exists \delta>0 such that \rho_2(f(y), f(x))<\epsilon whenever \rho_1(x, y)<\delta. In other words, f^{-1}(B(\epsilon, f(x)))\supset B(\delta, x). The map f is called continuous if it is continuous at each x\in X_1 and uniformly continuous if the \delta in the definition of continuity can be chosen independent of x.

Proposition. f: X_1\to X_2 is continuous iff f^{-1}(U) is open in X_1 for every open U\subset X_2.

Proof. If the latter condition holds, then for every x\in X_1 and \epsilon>0, the set f^{-1}(B(\epsilon, f(x))) is open and contains x, so it contains some ball about x; this means that f is continuous at x. Conversely, suppose that f is continuous and U is open in X_2. For each y\in U, \exists \epsilon_y>0 such that B(\epsilon_y, y)\subset U, and for each x\in f^{-1}(\{y\}), \exists\delta_x such that B(\delta_x, x)\subset f^{-1}(B(\epsilon_y, y))\subset f^{-1}(U). Thus f^{-1}(U)=\cup_{x\in f^{-1}}B(\delta_x, x) is open. $\Box$

“I understand now that the vulnerability I’ve always felt is the greatest strength a person can have. You can’t experience life without feeling life. What I’ve learned is that being vulnerable to somebody you love is not a weakness, it is strength.” ~ Elisabeth Shue

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 13-14.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.