Metric Spaces — Open and Closed Sets, Convergence

⚡ Key Takeaways

A metric space $(X,d)$ pairs a set $X$ with a distance function $d$ satisfying positivity, symmetry, and the triangle inequality. Open sets are unions of open balls, closed sets are their complements, and a sequence converges to a limit when its terms eventually stay within every ball around that limit. Used everywhere analysis lives — real numbers, function spaces, fractal geometry. Foundational framework crystallized by Maurice Fréchet (1906) and Felix Hausdorff (1914).

💡 The Core Insight

Distance generates topology. Once you can measure $d(x,y)$ , you can declare a point "interior" to a set if some ball around it stays inside, declare a set "open" if all its points are interior, and define a limit by "eventually inside every ball." Open and closed sets are not opposites in plain language — they are complement-linked, and a set can be both, neither, or one. Convergence and closedness are two faces of the same coin: closed sets are exactly those that contain all their sequential limits.

DefinitionMetric Space

A metric space is a pair $(X,d)$ where $X$ is a set and $d:X\times X\to[0,\infty)$ is a function — the metric — satisfying, for all $x,y,z\in X$ :

1

\quad d(x,y)=0 \iff x=y

Derivation hint: This is the "identity of indiscernibles" axiom; combined with nonnegativity it says distance vanishes exactly on the diagonal.

2

\quad d(x,y)=d(y,x)

Derivation hint: Symmetry — distance does not depend on direction of travel.

3

\quad d(x,z)\le d(x,y)+d(y,z)

Derivation hint: The triangle inequality; a detour through $y$ is never shorter than the direct route. Here $x,y,z$ are arbitrary points of $X$ . Nonnegativity ( $d\ge 0$ ) follows from the others: $0=d(x,x)\le d(x,y)+d(y,x)=2d(x,y)$ .

DefinitionOpen Ball

For $x_0\in X$ and radius $r>0$ , the open ball of radius $r$ centered at $x_0$ is

4

\quad B(x_0,r)=\{\,x\in X : d(x,x_0)<r\,\}

Derivation hint: The set of all points strictly closer than $r$ ; the boundary sphere is excluded by the strict inequality. The closed ball is $\bar{B}(x_0,r)=\{x:d(x,x_0)\le r\}$ .

DefinitionOpen and Closed Sets

A set $U\subseteq X$ is open if every point is an interior point:

5

\quad U \text{ open} \iff \forall x\in U,\ \exists r>0 \text{ such that } B(x,r)\subseteq U

Derivation hint: Openness = "room to wiggle"; around each member there is a whole ball still inside. A set $C\subseteq X$ is closed if its complement is open:

6

\quad C \text{ closed} \iff X\setminus C \text{ is open}

Derivation hint: Closedness is defined by complementation, not directly; this is the cleanest topological definition.

DefinitionInterior, Closure, Boundary

For $A\subseteq X$ :

7

\quad \operatorname{int}(A)=\{x\in A : \exists r>0,\ B(x,r)\subseteq A\}

Derivation hint: The largest open set contained in $A$ ; collect all interior points.

8

\quad \overline{A}=\{x\in X : \forall r>0,\ B(x,r)\cap A\neq\varnothing\}

Derivation hint: The closure adds all limit points; $x$ is in $\overline A$ iff every ball around it touches $A$ . It equals the smallest closed set containing $A$ .

9

\quad \partial A=\overline{A}\setminus\operatorname{int}(A)

Derivation hint: Boundary points have every ball meeting both $A$ and its complement.

DefinitionConvergence of a Sequence

A sequence $(x_n)_{n\ge1}$ in $X$ converges to $x\in X$ , written $x_n\to x$ , if

10

\quad \forall \varepsilon>0,\ \exists N\in\mathbb{N},\ \forall n\ge N:\ d(x_n,x)<\varepsilon

Derivation hint: Replace $|x_n-x|$ in the real-line definition by $d(x_n,x)$ ; "eventually within every ball." Equivalently $d(x_n,x)\to 0$ as a real sequence.

TheoremTopology Axioms from a Metric

Let $(X,d)$ be a metric space and $\mathcal{T}$ the collection of open sets. Then: 1. $\varnothing$ and $X$ are open. 2. Any union (arbitrary) of open sets is open. 3. Any finite intersection of open sets is open. Thus every metric space is a topological space.

ProofOpen sets form a topology

Step 1 (trivial sets). $\varnothing$ is open vacuously (no points to check). $X$ is open since any ball lies in $X$ .

Step 2 (arbitrary unions). Let $U=\bigcup_{i\in I}U_i$ with each $U_i$ open. Take $x\in U$ ; then $x\in U_{i_0}$ for some $i_0$ . By openness of $U_{i_0}$ there is $r>0$ with $B(x,r)\subseteq U_{i_0}\subseteq U$ . Hence $x$ is interior to $U$ , so $U$ is open.

Step 3 (finite intersections). Let $U=\bigcap_{k=1}^{n}U_k$ , each open. Take $x\in U$ . For each $k$ there is $r_k>0$ with $B(x,r_k)\subseteq U_k$ . Set $r=\min\{r_1,\dots,r_n\}>0$ (finiteness is essential here). Then $B(x,r)\subseteq B(x,r_k)\subseteq U_k$ for all $k$ , so $B(x,r)\subseteq U$ . Thus $U$ is open. $\blacksquare$

∎

TheoremSequential Characterization of Closed Sets

A set $C\subseteq X$ is closed if and only if it is sequentially closed: whenever $(x_n)\subseteq C$ and $x_n\to x$ in $X$ , then $x\in C$ .

ProofClosed = contains its limits

( $\Rightarrow$ ) Suppose $C$ is closed and $(x_n)\subseteq C$ with $x_n\to x$ . If $x\notin C$ then $x\in X\setminus C$ , which is open, so some $B(x,r)\subseteq X\setminus C$ . But $x_n\to x$ forces $x_n\in B(x,r)$ for large $n$ , contradicting $x_n\in C$ . Hence $x\in C$ .

( $\Leftarrow$ ) Suppose $C$ is sequentially closed; we show $X\setminus C$ is open. If not, some $x\in X\setminus C$ has no ball inside $X\setminus C$ : for every $n$ , $B(x,1/n)\cap C\neq\varnothing$ . Pick $x_n\in B(x,1/n)\cap C$ . Then $d(x_n,x)<1/n\to0$ , so $x_n\to x$ with $(x_n)\subseteq C$ . By hypothesis $x\in C$ , contradicting $x\in X\setminus C$ . Hence $X\setminus C$ is open and $C$ is closed. $\blacksquare$

∎

TheoremOpen Balls Are Open; Uniqueness of Limits

(a) Every open ball $B(x_0,r)$ is an open set. (b) In any metric space, sequential limits are unique: if $x_n\to x$ and $x_n\to y$ then $x=y$ .

ProofBalls open, limits unique

(a) Let $y\in B(x_0,r)$ , so $d(y,x_0)<r$ . Put $s=r-d(y,x_0)>0$ . For $z\in B(y,s)$ , the triangle inequality [3] gives $d(z,x_0)\le d(z,y)+d(y,x_0)<s+d(y,x_0)=r$ , so $z\in B(x_0,r)$ . Thus $B(y,s)\subseteq B(x_0,r)$ and the ball is open.

(b) Suppose $x_n\to x$ and $x_n\to y$ with $x\neq y$ , so $d(x,y)=\delta>0$ . Choose $N$ so that $n\ge N$ gives $d(x_n,x)<\delta/2$ and $d(x_n,y)<\delta/2$ . Then by [3], $\delta=d(x,y)\le d(x,x_n)+d(x_n,y)<\delta/2+\delta/2=\delta$ , a contradiction. So $x=y$ . $\blacksquare$

∎

mathematician · Maurice Fréchet | 1878–1973

In his 1906 doctoral thesis Sur quelques points du calcul fonctionnel, Fréchet introduced abstract spaces with a distance function — what he called classes (E) — generalizing convergence away from $\mathbb{R}^n$ to spaces of curves and functions. He recognized that limit and continuity needed only a notion of "écart" (gap), launching general topology and functional analysis. The very word metric space (espace métrique) and the axioms [1]–[3] descend directly from his work. Felix Hausdorff later (1914) refined and popularized these ideas in Grundzüge der Mengenlehre, embedding them in the broader axiomatic study of point-set topology.

🧠 Intuition

Think of $d$ as a ruler you carry everywhere. An open set is a country with no walls: standing anywhere inside, you can take a small step in any direction and remain in the country. A closed set may include its border fence — points on the edge belong to it. The number line illustrates the spectrum: $(0,1)$ is open (every point has wiggle room), $[0,1]$ is closed (it owns its endpoints), $[0,1)$ is neither, and $\varnothing$ and $\mathbb{R}$ are simultaneously both open and closed. Convergence is a "settling down" idea: no matter how tiny a target ball you draw around the limit, the sequence eventually crawls inside and never leaves. Crucially, which sets are open depends on the metric, but two different metrics can declare the same open sets (then they are topologically equivalent), in which case they agree on every convergence, continuity, and closedness question.

Figure

Open vs. closed: a point interior to an open set has a full ball inside, while a boundary point of a closed set sits on the rim

ExampleConvergence in the Euclidean plane

Given: $X=\mathbb{R}^2$ with Euclidean metric $d(p,q)=\sqrt{(p_1-q_1)^2+(p_2-q_2)^2}$ . Consider $x_n=\left(\dfrac{1}{n},\, 2-\dfrac{1}{n^2}\right)$ . Does it converge, and to what?

Step 1 — guess the limit. As $n\to\infty$ , $1/n\to0$ and $2-1/n^2\to2$ . Candidate limit $x=(0,2)$ .

Step 2 — compute the distance.

d(x_n,x)=\sqrt{\left(\tfrac1n\right)^2+\left(\tfrac1{n^2}\right)^2}=\sqrt{\tfrac1{n^2}+\tfrac1{n^4}}.

Step 3 — bound it. Since $1/n^4\le 1/n^2$ , we get $d(x_n,x)\le\sqrt{2/n^2}=\sqrt2/n$ .

Step 4 — apply the definition [10]. Given $\varepsilon>0$ , choose $N>\sqrt2/\varepsilon$ . Then $n\ge N\Rightarrow d(x_n,x)\le\sqrt2/n\le\sqrt2/N<\varepsilon$ .

Answer:

\boxed{x_n\to(0,2)\quad\text{in }(\mathbb{R}^2,d_{\text{Euclidean}})}

ExampleClosure and interior of a rational interval

Given: $X=\mathbb{R}$ with $d(x,y)=|x-y|$ . Let $A=(1,3]\cap\mathbb{Q}$ (rationals in the interval).

Step 1 — interior. Any open ball $B(x,r)=(x-r,x+r)$ contains irrationals, which are not in $A$ . So no point of $A$ has a ball inside $A$ :

\operatorname{int}(A)=\varnothing.

Step 2 — closure. Every ball around any $x\in[1,3]$ meets the rationals of $(1,3]$ (rationals are dense). Points outside $[1,3]$ have balls disjoint from $A$ .

\overline{A}=[1,3].

Step 3 — boundary. By [9], $\partial A=\overline A\setminus\operatorname{int}(A)=[1,3]\setminus\varnothing=[1,3]$ .

Answer:

\boxed{\operatorname{int}(A)=\varnothing,\quad \overline{A}=[1,3],\quad \partial A=[1,3]}

A set can have empty interior yet a "fat" closure — density makes the difference.

⚠️ Common mistake

"Closed means not open." False. Open and closed are not complementary properties. In any metric space $\varnothing$ and $X$ are both open and closed (clopen). The half-open interval $[0,1)$ is neither. The correct relationship is via complements [6]: $C$ is closed iff $X\setminus C$ is open.

"Infinite intersections of open sets are open." False. In $\mathbb{R}$ , $\bigcap_{n\ge1}(-1/n,\,1/n)=\{0\}$ , a closed set, not open. Only finite intersections are guaranteed open (see [Step 3] of the topology proof).

"The closed ball equals the closure of the open ball." Not always! In a general metric space $\overline{B(x,r)}\subseteq \bar B(x,r)$ , but equality can fail.

CounterexampleClosed ball strictly larger than closure of open ball

Take $X$ to be any set with two or more points equipped with the discrete metric

d(x,y)=\begin{cases}0,& x=y\\ 1,& x\neq y.\end{cases}

Fix a point $p$ and radius $r=1$ .

Open ball: $B(p,1)=\{x:d(x,p)<1\}=\{p\}$ . Its closure is $\overline{\{p\}}=\{p\}$ (singletons are closed here).
Closed ball: $\bar B(p,1)=\{x:d(x,p)\le1\}=X$ (everything is within distance $1$ ).

So $\overline{B(p,1)}=\{p\}\subsetneq X=\bar B(p,1)$ . The naive intuition from $\mathbb{R}^n$ breaks. (In the discrete metric every subset is both open and closed, and a sequence converges iff it is eventually constant.)

connection

The open/closed/convergence machinery is the gateway to the rest of analysis. Continuity is characterized cleanly here: $f$ is continuous iff preimages of open sets are open, iff $x_n\to x \Rightarrow f(x_n)\to f(x)$ — see Continuous Functions on Metric Spaces. Completeness asks whether Cauchy sequences converge, refining the convergence notion of [10] into Complete Metric Spaces and powering the Banach Fixed-Point Theorem. Compactness (Compact Metric Spaces) combines closedness with a boundedness-like finiteness via sequential or open-cover formulations. The whole framework abstracts to Topological Spaces, where open sets are taken as primitive and the metric is forgotten. Finally, different metrics yielding the same open sets connect to Equivalent Metrics and Norms.

🔬 Where this appears

⏳ History

- 1817 — Bolzano gives an early rigorous notion of continuity and limit, foreshadowing the

\varepsilon

-language.

- 1872 — Weierstrass and Heine codify the

\varepsilon

–

\delta

/

\varepsilon

–

N

definitions of continuity and convergence on

\mathbb{R}

.

- 1872 — Cantor and Dedekind construct the reals; Cantor introduces accumulation (limit) points and derived sets, the seed of "closure."

- 1906 — Maurice Fréchet's thesis introduces abstract metric spaces (classes (E)) and the axioms of distance, generalizing convergence to function spaces.

- 1910 — Fréchet and F. Riesz develop early functional-analytic spaces; the notion of completeness in abstract settings emerges.

- 1914 — Felix Hausdorff's Grundzüge der Mengenlehre axiomatizes neighborhoods and topological spaces, placing metric spaces inside a wider theory; the Hausdorff separation axiom is born.

- 1922 — Stefan Banach's thesis founds normed/complete spaces (Banach spaces) and proves the contraction mapping theorem.

- 1906–1930 — The Moscow and Polish (Lwów–Warsaw) schools (Urysohn, Alexandroff, Kuratowski, Sierpiński) systematize open/closed sets, closure operators (Kuratowski axioms), and metrization theorems.

- 1933 — Urysohn's and later metrization theorems characterize exactly which topological spaces arise from a metric, closing the abstraction loop.

open-problem

While metric spaces are well-understood, their boundaries spark ongoing research. Metrizability: precisely which topological spaces admit a compatible metric is governed by the Nagata–Smirnov and Bing metrization theorems, but generalizations to non-Hausdorff or quasi-metric settings remain subtle. Coarse geometry & large-scale structure: Gromov's program studies metric spaces "up to bounded distortion," central to geometric group theory and still generating open questions about embeddings of groups into Hilbert space (the Novikov and coarse Baum–Connes conjectures). Metric embedding theory (active in theoretical computer science) asks how faithfully arbitrary finite metrics embed into $\ell^p$ spaces with low distortion — Bourgain's theorem gives $O(\log n)$ , and optimal distortions for specific families stay open. Wasserstein / optimal-transport metrics on probability measures are an explosively active area with deep open questions in geometry and PDE. Even the innocent fact that closed balls can differ from closures of open balls signals how much geometric information a bare metric retains — and how much it discards.

Metric Spaces — Open and Closed Sets, Convergence

Formal Statement

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also