Metric Spaces and the Topology of the Real Line

Chapter 1 built $\mathbb{R}$ from its order and completeness. This chapter extracts a different piece of structure — distance — and studies what can be said using distance alone, in a setting general enough to include $\mathbb{R}^n$, function spaces and the discrete two-point space on equal footing. The resulting notions of open set, closed set, limit point and convergence are the vocabulary for everything from here on: continuity (defined via open sets, not just $\varepsilon$–$\delta$), compactness (Chapter 3), and completeness in the general sense (Chapter 4).

Have to hand: the triangle inequality for $|\cdot|$ on $\mathbb{R}$, and comfort with quantifiers. We work mostly in $\mathbb{R}$ with its usual metric $d(x,y)=|x-y|$, but state every definition for a general metric space so that the pattern is visible from the start.

Why generalise at all? Once a theorem is proved using only the metric axioms (not the specific formula $|x-y|$), it is automatically true in $\mathbb{R}^n$, in spaces of continuous functions, and anywhere else a metric is defined — without re-proving it. This is the entire payoff of the abstraction.

1. Metric Spaces: Definition and Examples

Definition. A metric on a set $X$ is a function $d : X\times X \to \mathbb{R}$ satisfying, for all $x,y,z \in X$:

1. $d(x,y) \ge 0$, with $d(x,y) = 0$ if and only if $x = y$ (positivity);
2. $d(x,y) = d(y,x)$ (symmetry);
3. $d(x,z) \le d(x,y) + d(y,z)$ (triangle inequality).

The pair $(X,d)$ is a metric space.

Examples.

• $\mathbb{R}$ with $d(x,y) = |x-y|$: the triangle inequality here is the familiar $|x-z| \le |x-y|+|y-z|$.
• $\mathbb{R}^n$ with the Euclidean metric $d(\vec{x},\vec{y}) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}$.
• Any set $X$ with the discrete metric $d(x,y) = 0$ if $x=y$ and $d(x,y)=1$ otherwise. All three axioms are checked directly from the definition; this example shows a metric need carry no "geometric" content at all.

2. Open Balls and Open Sets

Definition. For $x\in X$ and $r>0$, the open ball of radius $r$ about $x$ is $B(x,r) = \{y \in X : d(x,y) < r\}$. A set $U \subseteq X$ is open if for every $x\in U$ there exists $r>0$ with $B(x,r)\subseteq U$.

In $\mathbb{R}$, $B(x,r) = (x-r,x+r)$, so "open" recovers the familiar picture: every point of $U$ has some room around it still inside $U$.

Worked example. Every open interval $(a,b)$ is open in $\mathbb{R}$. Given $x\in(a,b)$, let $r = \min(x-a,\,b-x) > 0$. If $y\in B(x,r)$ then $|y-x|<r\le x-a$ gives $y > x - (x-a) = a$, and $|y-x|<r\le b-x$ gives $y < x+(b-x) = b$; so $B(x,r)\subseteq(a,b)$. Since $x$ was arbitrary, $(a,b)$ is open.

Theorem. (i) $\emptyset$ and $X$ are open. (ii) The union of any collection of open sets is open. (iii) The intersection of finitely many open sets is open.

Proof. (i) is immediate ($\emptyset$ vacuously; $X$ since $B(x,r)\subseteq X$ always). (ii) Let $\{U_\alpha\}_{\alpha\in A}$ be open and $x\in \bigcup_\alpha U_\alpha$; then $x\in U_{\alpha_0}$ for some $\alpha_0$, and openness of $U_{\alpha_0}$ gives $r>0$ with $B(x,r)\subseteq U_{\alpha_0}\subseteq\bigcup_\alpha U_\alpha$. (iii) Let $U_1,\dots,U_n$ be open and $x\in\bigcap_{i=1}^n U_i$. For each $i$ there is $r_i>0$ with $B(x,r_i)\subseteq U_i$; take $r=\min(r_1,\dots,r_n)>0$ (a minimum of finitely many positive numbers is positive). Then $B(x,r)\subseteq B(x,r_i)\subseteq U_i$ for every $i$, so $B(x,r)\subseteq\bigcap_i U_i$. $\blacksquare$

Key distinction. Finiteness in (iii) is essential: $\bigcap_{n=1}^{\infty}\left(-\frac1n,\frac1n\right) = \{0\}$, an intersection of open sets that is not open (no ball around $0$ fits inside $\{0\}$). Infinite intersections of open sets can fail to be open.

3. Closed Sets

Definition. $F\subseteq X$ is closed if its complement $X\setminus F$ is open.

Theorem (De Morgan dual of the open-set theorem). (i) $\emptyset$ and $X$ are closed. (ii) The intersection of any collection of closed sets is closed. (iii) The union of finitely many closed sets is closed.

Proof. Apply De Morgan's laws to the open-set theorem: $X\setminus\bigcap_\alpha F_\alpha = \bigcup_\alpha(X\setminus F_\alpha)$ is a union of open sets, hence open, so $\bigcap_\alpha F_\alpha$ is closed; similarly $X\setminus\bigcup_{i=1}^n F_i = \bigcap_{i=1}^n (X\setminus F_i)$ is a finite intersection of open sets, hence open. $\blacksquare$

Closed and open are not opposites in the everyday sense: a set can be both ($\emptyset$ and $X$ always are), and a set can be neither, e.g. $[0,1)\subseteq\mathbb{R}$ is neither open (no ball around $0$ fits inside) nor closed (its complement $(-\infty,0)\cup[1,\infty)$ is not open, since no ball around $1$ fits inside the complement).

4. Limit Points and Closure

Definition. $x\in X$ is a limit point (accumulation point) of $S\subseteq X$ if every open ball $B(x,r)$, $r>0$, contains a point of $S$ other than $x$ itself. A point $s\in S$ that is not a limit point of $S$ is an isolated point of $S$.

Note a limit point of $S$ need not belong to $S$. The closure $\overline{S} = S \cup \{\text{limit points of } S\}$.

Theorem. $\overline S$ is the smallest closed set containing $S$; equivalently, $\overline S = \bigcap \{F : F \text{ closed},\ S\subseteq F\}$.

Proof sketch. First, $\overline{S}$ is closed: if $x\notin\overline S$, then $x\notin S$ and $x$ is not a limit point of $S$, so some $B(x,r)$ contains no point of $S$ other than $x$ — but $x\notin S$, so $B(x,r)\cap S=\emptyset$. For $y\in B(x,r)$, openness of balls gives a smaller ball around $y$ still inside $B(x,r)$, hence still missing $S$, so $y\notin\overline S$ either (no point of that smaller ball lies in $S$, ruling out both membership and limit-point status). Thus $B(x,r)\subseteq X\setminus\overline S$, proving $X\setminus \overline S$ open. Second, any closed $F\supseteq S$ must contain every limit point of $S$ (if $x$ were a limit point of $S$ outside $F$, the open set $X\setminus F$ would be a ball-free neighbourhood of $x$ missing $S$ entirely near $x$, contradicting $x$ being a limit point), so $F \supseteq \overline S$. Hence $\overline S$ is the least closed superset. $\blacksquare$

Worked example. $\overline{\mathbb{Q}} = \mathbb{R}$. Every $x\in\mathbb{R}$ is a limit point of $\mathbb{Q}$: given $r>0$, density of $\mathbb{Q}$ (Chapter 1) gives a rational in $(x-r,x+r)\setminus\{x\}$ unless $x$ itself is rational, in which case density applied to $(x, x+r)$ still gives a rational distinct from $x$ inside $B(x,r)$. So every real is in $\mathbb{Q}$ or a limit point of it, giving $\overline{\mathbb{Q}}=\mathbb{R}$ — a set can have closure equal to the entire space while being "small" in other senses (countable, as here).

5. Interior and Boundary

Definition. The interior $S^{\circ}$ is the set of $x\in S$ for which some $B(x,r)\subseteq S$. The boundary $\partial S = \overline S \setminus S^{\circ}$.

$S$ is open iff $S = S^{\circ}$; $S$ is closed iff $S = \overline S$. For $S=(a,b]$ in $\mathbb{R}$: $S^{\circ}=(a,b)$ (the point $b$ has no ball of any radius fitting inside $S$, since points just above $b$ are excluded), $\overline S = [a,b]$, and $\partial S = \{a,b\}$.

6. Convergence of Sequences in a Metric Space

Definition. A sequence $(x_n)$ in $(X,d)$ converges to $x\in X$, written $x_n \to x$, if for every $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $d(x_n,x)<\varepsilon$ for all $n\ge N$. Equivalently, every open ball $B(x,\varepsilon)$ contains all but finitely many terms of the sequence.

Theorem (uniqueness of limits). If $x_n\to x$ and $x_n\to x'$, then $x=x'$.

Proof. Suppose $x\ne x'$; let $\varepsilon = d(x,x')/2 > 0$ (positive by the positivity axiom, since $x\ne x'$). Choose $N_1$ with $d(x_n,x)<\varepsilon$ for $n\ge N_1$, and $N_2$ with $d(x_n,x')<\varepsilon$ for $n\ge N_2$. For $n\ge\max(N_1,N_2)$, the triangle inequality gives $d(x,x') \le d(x,x_n)+d(x_n,x') < \varepsilon+\varepsilon = d(x,x')$, a contradiction. $\blacksquare$

Theorem. $x$ is a limit point of $S$ if and only if there is a sequence $(x_n)$ in $S\setminus\{x\}$ with $x_n\to x$.

Proof sketch. ($\Leftarrow$) If such a sequence exists, every $B(x,r)$ eventually contains terms $x_n\ne x$ by convergence, so it contains a point of $S$ other than $x$. ($\Rightarrow$) If $x$ is a limit point, for each $n\in\mathbb{N}$ apply the definition with $r=1/n$ to choose $x_n\in B(x,1/n)\cap S$, $x_n\ne x$; then $d(x_n,x)<1/n\to0$, so $x_n\to x$ by the Archimedean property. $\blacksquare$

This reformulation is why many topology proofs in this course can be carried out with sequences, and will be used constantly from Chapter 3 onward.

7. Equivalent Metrics

Two metrics $d_1,d_2$ on $X$ are (topologically) equivalent if they generate the same open sets. This can happen even when $d_1\ne d_2$ as functions: on $\mathbb{R}^n$, the Euclidean metric and the "taxicab" metric $d_1(\vec x,\vec y)=\sum_i|x_i-y_i|$ generate the same open sets, because each metric ball contains and is contained in a ball of the other metric (with different radii) — a fact used freely in later multivariable chapters to pass between coordinate estimates and geometric ones without comment.

Common Exam Pitfalls

• Writing "closed" to mean "not open". These are independent properties: $[0,1)$ is neither, and $\emptyset,X$ are both. Always check the actual definitions, never infer one from the failure of the other.
• Forgetting finiteness in the intersection-of-opens theorem. $\bigcap_n(-1/n,1/n)=\{0\}$ is the standard counterexample when infinitely many open sets are intersected.
• Confusing limit point with element of the set. A limit point of $S$ need not lie in $S$ (e.g. every point of $\overline{\mathbb Q}\setminus\mathbb Q$ is a limit point of $\mathbb Q$ not in $\mathbb Q$); conversely an element of $S$ need not be a limit point (isolated points).
• Asserting $\overline{S} = S$ whenever $S$ "looks closed". Always verify via the actual limit-point definition or by checking the complement is open; intuition about "edges" can mislead for unusual sets like $\mathbb{Q}$.
• Treating metric equivalence as requiring $d_1=d_2$. Equivalent metrics only need to produce the same open sets; the distance values themselves can differ substantially (even by unbounded ratios in general, though not in the taxicab/Euclidean example).
• Dropping symmetry or the strict "iff" in positivity when verifying a candidate is a metric. All three axioms must be checked; a function satisfying symmetry, the triangle inequality and $d(x,x)=0$ (without ruling out $d(x,y)=0$ for $x\ne y$) is a pseudometric, not a metric.

Summary

• A metric space $(X,d)$ satisfies positivity ($d(x,y)=0\iff x=y$), symmetry, and the triangle inequality.
• Open ball: $B(x,r)=\{y:d(x,y)<r\}$; open set: every point has a ball fitting inside it.
• Arbitrary unions and finite intersections of open sets are open; by complementation, arbitrary intersections and finite unions of closed sets are closed.
• Limit point of $S$: every ball around it meets $S$ at a point other than itself; closure $\overline S = S\cup\{\text{limit points}\}$ is the smallest closed set containing $S$.
• Interior $S^\circ$: points with a ball entirely inside $S$; boundary $\partial S = \overline S\setminus S^\circ$.
• Convergence: $x_n\to x$ iff $\forall\varepsilon>0\ \exists N:\ n\ge N\Rightarrow d(x_n,x)<\varepsilon$; limits are unique, proved via the triangle inequality.
• $x$ is a limit point of $S$ iff some sequence in $S\setminus\{x\}$ converges to $x$ — the bridge between topological and sequential language used throughout the rest of the course.
• Equivalent metrics generate the same open sets without needing to be numerically equal.