Metric Spaces

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property), Ordered field) constructed in this library (The real numbers) and carrying its order (Order on the reals).

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

• (M1) Separation. $d(x,y) = 0$ if and only if $x = y$.
• (M2) Symmetry. $d(x,y) = d(y,x)$.
• (M3) Triangle inequality. $d(x,z) \le d(x,y) + d(y,z)$.

A metric space is a pair $(X,d)$ consisting of a set $X$ and a metric $d$ on it. The elements of $X$ are its points and $d(x,y)$ is the distance from $x$ to $y$. When only one metric is in play we write $X$ for $(X,d)$; when several are, the metric is always named.

The values of a metric are real numbers. The codomain is $\mathbb{R}$, so $d(x,y)$ is an honest element of the complete ordered field and every inequality above is an inequality there. No infinite value is permitted; Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here records why extended metrics are not treated in this library.

Nonnegativity is deliberately absent from the axiom list. Many texts add a fourth axiom $d(x,y) \ge 0$. It is redundant: (M1), (M2) and (M3) already force it, as Nonnegativity of a metric is a consequence of the other axioms, not an axiom proves. Nothing below assumes it before that lemma is available.

Pseudometric. A pseudometric on $X$ is a function $p : X \times X \to \mathbb{R}$ satisfying (M2), (M3) and the weakening

• (M1') Reflexivity. $p(x,x) = 0$ for every $x \in X$

of (M1). A pseudometric may therefore assign distance $0$ to two distinct points. Every metric is a pseudometric, and a pseudometric is a metric exactly when $p(x,y) = 0$ forces $x = y$.

Ultrametric. An ultrametric on $X$ is a metric $d$ that in addition satisfies

• (M3') Strong triangle inequality. $d(x,z) \le \max\{d(x,y), d(y,z)\}$

for all $x, y, z \in X$, where the maximum is that of a two-element subset of $\mathbb{R}$, which exists and is one of the two elements (Maximum and minimum of a set, Every nonempty finite set of reals has a maximum and a minimum). An ultrametric space is a pair $(X,d)$ with $d$ an ultrametric.

• (M3') is a genuine strengthening of (M3), not an independent axiom on top of it. A function satisfying (M1), (M2) and (M3') automatically satisfies (M3): by Nonnegativity of a metric is a consequence of the other axioms, not an axiom such a function is nonnegative, and for nonnegative reals $a, b$ one has $\max\{a,b\} \le a + b$, since the maximum is one of $a, b$ and the other summand is $\ge 0$. So "a metric satisfying (M3')" and "a function satisfying (M1), (M2), (M3')" describe the same objects, and the definition above may be read either way.

• Why the biconditional form of (M1). Splitting (M1) into "$d(x,x) = 0$" and "$d(x,y) = 0 \Rightarrow x = y$" gives the same notion; the split form is what makes the pseudometric weakening above a matter of deleting one clause. The naming fork between pseudometric and semimetric, which is live in the literature, is settled for this library in Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here.

• The metric is part of the data. Two different metrics on the same set are two different metric spaces, even when they have the same open sets. That is why Topologically, uniformly and Lipschitz equivalent metrics on a set compares metrics at three separate strengths rather than one, and why a property can be invariant under one of them and not under another (FALSE: boundedness of a metric space is determined by its topology).

Let $X$ be a set and let $p : X \times X \to \mathbb{R}$ satisfy the reflexivity axiom (M1') $p(x,x) = 0$ and the symmetry axiom (M2) $p(x,y) = p(y,x)$ of Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric. Then:

1. If $p$ satisfies the triangle inequality (M3), then $p(x,y) \ge 0$ for all $x, y \in X$.
2. If $p$ satisfies the strong triangle inequality (M3'), then $p(x,y) \ge 0$ for all $x, y \in X$.

In particular every metric, every pseudometric and every ultrametric (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) takes only nonnegative values. Nonnegativity is therefore a theorem about the axiom list this library uses, not a fourth axiom, and no statement on this page needs to assume it separately.

• What each claim uses. Claim 1 is the familiar two-line argument $0 = p(x,x) \le p(x,y) + p(y,x) = 2p(x,y)$ followed by the observation that a negative real added to itself stays negative. Claim 2 does not need that second half at all: the strong triangle inequality delivers $0 \le p(x,y)$ in one step, because the maximum of a real number with itself is that number.
• Symmetry is used in both claims and cannot be dropped. Without (M2) the instantiation of step 1.1 only gives $0 \le p(x,y) + p(y,x)$, which leaves the possibility that one of the two values is negative and the other larger and positive. Dropping (M2) instead of weakening (M1) gives the notion usually called a quasimetric, which this library does not treat; for it the argument above is unavailable, so nonnegativity is not redundant there and is imposed as part of the definition (Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here).

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $x \in X$ and let $r \in \mathbb{R}$ with $r > 0$ (Order on the reals). Define

$B(x,r) := \{\, y \in X : d(x,y) < r \,\}, \qquad \bar B(x,r) := \{\, y \in X : d(x,y) \le r \,\}, \qquad S(x,r) := \{\, y \in X : d(x,y) = r \,\}.$

$B(x,r)$ is the open ball, $\bar B(x,r)$ the closed ball and $S(x,r)$ the sphere of centre $x$ and radius $r$. The radius is always a strictly positive real; a ball of radius $0$ or of negative radius is never written in this library.

Immediate consequences of the definitions. For every $x \in X$ and $r > 0$:

• $x \in B(x,r)$, because $d(x,x) = 0 < r$ (axiom (M1) of Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric); in particular open and closed balls are nonempty.
• $B(x,r) \subseteq \bar B(x,r)$ and $S(x,r) \subseteq \bar B(x,r)$, and $\bar B(x,r)$ is the disjoint union of $B(x,r)$ and $S(x,r)$, by trichotomy of the order of $\mathbb{R}$ (Complete ordered field (least-upper-bound property), Ordered field): each $y$ satisfies exactly one of $d(x,y) < r$, $d(x,y) = r$, $d(x,y) > r$.
• If $0 < s \le r$ then $B(x,s) \subseteq B(x,r)$ and $\bar B(x,s) \subseteq \bar B(x,r)$, by transitivity of the order.
• Nonnegativity of the metric (Nonnegativity of a metric is a consequence of the other axioms, not an axiom) is what forces the radius convention, and it forces it for the open ball only: if $r \le 0$ then $B(x,r) = \{y : d(x,y) < r\}$ is empty, because $d(x,y) \ge 0 \ge r$ for every $y$. The other two sets behave differently at $r = 0$, and the convention $r > 0$ excludes them for uniformity rather than for emptiness: $\bar B(x,0) = S(x,0) = \{x\}$, since $d(x,y) \le 0$ together with $d(x,y) \ge 0$ gives $d(x,y) = 0$ and hence $y = x$ by (M1). For $r < 0$ all three sets are empty.

A sphere may be empty, and so the three sets are not on a par. For $r > 0$ the open and closed balls always contain $x$, but nothing in the definition produces a point at distance exactly $r$ from $x$. If a metric takes only the values $0$ and $1$, as the discrete metric on the companion page does, then $S(x,2) = \emptyset$ while $B(x,2) = \bar B(x,2)$ is the whole space. So nonemptiness of a sphere is never available by convention: where it is used, it is proved.

The ambient space is part of the notation. $B(x,r)$ depends on $(X,d)$ and not on $x$ and $r$ alone. When more than one space or more than one metric is in play we write $B_X(x,r)$, or $B_d(x,r)$, and likewise for $\bar B$ and $S$. This matters as soon as subspaces appear (Isometry, isometric embedding, and the subspace metric on a subset): a ball of a subspace is the trace on it of a ball of the ambient space, and the two are different sets.

• The names "open ball" and "closed ball" are justified, not merely suggestive. That $B(x,r)$ is an open set and $\bar B(x,r)$ a closed set in the metric topology is proved in Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed; the words are used here only as names for the three sets displayed above.
• The closed ball is not in general the closure of the open ball, and the sphere is not in general the boundary of either. Both failures are recorded on this page as FALSE: in every metric space the closure of $B(x,r)$ is the closed ball of radius $r$ and witnessed on the companion page. The safe reading of the three names is the displayed one and nothing more.

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and let $A, B \subseteq X$.

Bounded subset. $A$ is bounded if $A = \emptyset$ or there are $x_0 \in X$ and a real $r > 0$ with $A \subseteq B(x_0, r)$ (Open ball, closed ball and sphere in a metric space). The space $(X,d)$ is a bounded metric space if $X$ is a bounded subset of itself.

Diameter, for nonempty bounded $A$ only. Suppose $A$ is nonempty and bounded, and put

$D(A) := \{\, d(a,b) : a, b \in A \,\} \subseteq \mathbb{R}.$

Then $D(A)$ is nonempty, since $A$ is, and it is bounded above: fixing $x_0$ and $r$ with $A \subseteq B(x_0,r)$, every $a, b \in A$ satisfy $d(a,b) \le d(a,x_0) + d(x_0,b) < r + r$ by the triangle inequality, symmetry (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and addition of inequalities (Order is preserved by adding a constant and by adding inequalities, Ordered field), so $r + r$ is an upper bound of $D(A)$ (Lower bound, bounded below, bounded set). Hence $D(A)$ has a least upper bound in $\mathbb{R}$ by the least-upper-bound property (Complete ordered field (least-upper-bound property)), and that bound is unique (Suprema and infima are unique). Define

$\operatorname{diam}(A) := \sup D(A).$

Distance from a point to a set, for nonempty $A$ only. Let $x \in X$ and let $A$ be nonempty, and put $E(x,A) := \{\, d(x,a) : a \in A \,\}$. Then $E(x,A)$ is nonempty and bounded below by $0$, since a metric is nonnegative (Nonnegativity of a metric is a consequence of the other axioms, not an axiom, Lower bound, bounded below, bounded set), so it has a greatest lower bound (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)), unique by Suprema and infima are unique. Define

$d(x,A) := \inf E(x,A).$

Distance between two sets, for nonempty $A$ and $B$ only. Put $E(A,B) := \{\, d(a,b) : a \in A,\ b \in B \,\}$, again nonempty and bounded below by $0$, and define

$d(A,B) := \inf E(A,B).$

Every one of the three scope restrictions is load bearing. In this library $\sup$ and $\inf$ denote real numbers and are written only after existence has been established; the extended real line is introduced on a later page and is not used for the suprema and infima taken here, and no convention $\sup \emptyset = -\infty$ is in force in this development (Conventions: $\sup \emptyset$, unbounded sets, and the extended reals). Accordingly:

• $\operatorname{diam}(A)$ is defined exactly when $A$ is nonempty and bounded. It is not defined for $A = \emptyset$, and it is not defined, not even as an infinite value, for an unbounded $A$.
• $d(x,A)$ is defined exactly when $A \ne \emptyset$, and $d(A,B)$ exactly when both $A$ and $B$ are nonempty. No boundedness is needed for these two, because $0$ is always a lower bound.

• Diameter and the distance functions are nonnegative. For nonempty bounded $A$ and any $a \in A$ we have $0 = d(a,a) \in D(A)$, so $\operatorname{diam}(A) \ge 0$; and $d(x,A) \ge 0$, $d(A,B) \ge 0$ because $0$ is a lower bound of the sets they are infima of (Nonnegativity of a metric is a consequence of the other axioms, not an axiom, Greatest lower bound (infimum)).
• $d(A,B)$ is not a metric on the nonempty subsets of $X$. It is symmetric and vanishes on $A = B$, but two distinct disjoint sets can be at distance $0$, so the separation axiom (M1) of Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric fails; the witness is on the companion page. The letter $d$ is reused for three different functions here, the metric, the point-to-set distance and the set-to-set distance, only because the arguments make the intended one unambiguous.
• $d(x,A)$ is the special case $d(\{x\},A)$, since $E(\{x\},A) = E(x,A)$, and the two infima therefore agree by uniqueness (Suprema and infima are unique).

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and let $x, y, z \in X$. Then

$|d(x,z) - d(y,z)| \le d(x,y),$

where $|\cdot|$ is the absolute value of $\mathbb{R}$ (Absolute value in an ordered field).

• Read with $z$ fixed, this says the function $u \mapsto d(u,z)$ does not increase distances: its values at $x$ and at $y$ differ by at most $d(x,y)$. That is the model for $|d(x,A) - d(y,A)| \le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz, which proves the same estimate with the point $z$ replaced by a nonempty set.
• The inequality specialises, on $\mathbb{R}$ with $d(u,v) = |u-v|$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded), to the familiar $\big||x - z| - |y - z|\big| \le |x - y|$.

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $A \subseteq X$ be nonempty and let $x, y \in X$. Then

$|d(x,A) - d(y,A)| \le d(x,y),$

with $d(\cdot,A)$ the distance to a nonempty set (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space). Thus the real-valued function $u \mapsto d(u,A)$ changes by at most $d(u,v)$ between $u$ and $v$: it is $1$-Lipschitz.

• Where the nonemptiness is used. Twice, and both times essentially: it is what makes $d(x,A)$ and $d(y,A)$ exist at all (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space), and it is what makes $E(x)$ and $E(y)$ nonempty so that "greatest lower bound" has content. For $A = \emptyset$ the statement has no meaning in this library, since $\inf \emptyset$ is undefined.
• The point case is The reverse triangle inequality $|d(x,z) - d(y,z)| \le d(x,y)$ in any metric space: taking $A = \{z\}$ gives $E(u) = \{d(u,z)\}$, whose infimum is $d(u,z)$, and the conclusion becomes $|d(x,z) - d(y,z)| \le d(x,y)$.
• The constant $1$ is best possible in general: on $\mathbb{R}$ with $A = \{0\}$ the function is $u \mapsto |u|$, and $\big||x| - |y|\big| = |x-y|$ whenever $x$ and $y$ have the same sign.

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric).

A subset $U \subseteq X$ is open in $(X,d)$ if for every $x \in U$ there is a real $r > 0$ with $B(x,r) \subseteq U$ (Open ball, closed ball and sphere in a metric space). A subset $F \subseteq X$ is closed in $(X,d)$ if its complement $X \setminus F$ is open.

The collection

$\mathcal{T}_d := \{\, U \subseteq X : U \text{ is open in } (X,d) \,\}$

of all open subsets is the metric topology of $d$ on $X$. A subset of $X$ that is both open and closed is called clopen.

Two sets are open for trivial reasons. $\emptyset$ is open, because the defining condition quantifies over no points; and $X$ is open, because $B(x,r) \subseteq X$ for every $x$ and every $r > 0$. Consequently $X$ and $\emptyset$ are also closed, and both are clopen.

A neighbourhood of a point $x$ is any open set containing $x$. The condition above therefore reads: $U$ is open exactly when every point of $U$ has a ball around it inside $U$, and it is the balls alone that have to be tested.

The metric, not the set, determines $\mathcal{T}_d$. Two metrics on the same set may have different metric topologies, and two different metrics may have the same one; the systematic comparison is Topologically, uniformly and Lipschitz equivalent metrics on a set.

• What "topology" means here. $\mathcal{T}_d$ is defined above as a collection of subsets of $X$; the abstract notion of a topological space, a collection of subsets closed under arbitrary unions and finite intersections taken as primitive data, is introduced on a later page and is not used here. What is proved here is that $\mathcal{T}_d$ has exactly those closure properties (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed), which is what licenses the word.
• Open and closed are not opposites. A set may be neither ($[0,1)$ inside $\mathbb{R}$, once the usual metric is available from The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded) or both ($\emptyset$ and $X$ always, and in some spaces every subset at once, as the discrete metric on the companion page shows). "Not open" is never a synonym for "closed".
• Closedness is complementation, and nothing else, at this stage. The description of closed sets by limits of sequences, and the description of the closure as an infimum of distances, are theorems proved later on this page (The closure of a nonempty $A$ is $\{x : d(x,A) = 0\}$, equals $A$ together with its limit points, and is the smallest closed superset, A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed), not part of the definition.

Define $d_{\mathbb{R}} : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ by $d_{\mathbb{R}}(x,y) := |x - y|$ (Absolute value in an ordered field). Then:

1. $d_{\mathbb{R}}$ is a metric on $\mathbb{R}$ (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric); it is called the usual metric of $\mathbb{R}$.
2. For $x \in \mathbb{R}$ and $r > 0$ the open ball is the bounded open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Open ball, closed ball and sphere in a metric space) $B(x,r) = (x-r,\ x+r),$ and the closed ball is $\bar B(x,r) = [x-r,\ x+r]$.
3. Consequently $U \subseteq \mathbb{R}$ is open in the metric topology of $d_{\mathbb{R}}$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) exactly when for every $x \in U$ there is $r > 0$ with $(x-r, x+r) \subseteq U$. This topology is called the usual topology of $\mathbb{R}$.
4. $(\mathbb{R}, d_{\mathbb{R}})$ is not a bounded metric space (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space): no ball contains $\mathbb{R}$, so $\operatorname{diam}(\mathbb{R})$ is not defined.