Analysis Foundations: Metric Spaces, Order, and Limits

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The shape of an open ball depends on the metric

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Radius r

2.0

Open ball Bᵣ(p) in ℝ²

Bᵣ(p) = { x ∈ X : d(x, p) < r }

How to use this reference

This is a compact glossary across several parts of introductory analysis, not a chapter limited to metric spaces. It moves from algebraic language to distance and topology, then to order, completeness, and limits. The examples in $\mathbb R$ provide a common setting; later results use these ideas in different combinations.

Definitions

Algebraic language

1. Rings, fields, and distributive laws

A ring is a set $R$ equipped with two binary operations, addition $+$ and multiplication $\cdot$, such that:

1. $(R,+)$ is an abelian group.
2. Multiplication is associative: $(xy)z=x(yz)$ for all $x,y,z\in R$.
3. Multiplication distributes over addition on both sides: $x(y+z)=xy+xz$ and $(x+y)z=xz+yz$ for all $x,y,z\in R$.

A ring is commutative if $xy=yx$ for all $x,y\in R$. Conventions differ on whether a ring is required to have a multiplicative identity; here that requirement is stated explicitly when needed. A field is a commutative ring with an identity $1\ne 0$ in which every nonzero element has a multiplicative inverse. Thus, for each $x\ne0$, there is $y$ such that $xy=yx=1$.

Distance and topology

2. Metric

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

1. Nonnegativity: $d(x,y)\ge0$.
2. Identity of indiscernibles: $d(x,y)=0$ if and only if $x=y$.
3. Symmetry: $d(x,y)=d(y,x)$.
4. Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$.

A pseudometric satisfies the same conditions except that identity of indiscernibles is weakened to $d(x,y)=0$ being possible for distinct points. (Some texts use semimetric for related but differing notions, so the term is not used here.)

3. Metric space

A metric space is a pair $(X,d)$ consisting of a set $X$ and a metric $d$ on it.

4. Euclidean distance

For a positive integer $n$, the Euclidean space $\mathbb R^n$ consists of vectors $x=(x_1,\ldots,x_n)$ with real coordinates. Its standard metric is

$d(x,y)=\left(\sum_{i=1}^n |x_i-y_i|^2\right)^{1/2}.$

5. Open ball

Given a metric space $(X,d)$, a center $x_0\in X$, and a radius $r>0$, the open ball of radius $r$ centered at $x_0$ is

$B_r(x_0)=\{x\in X:d(x_0,x)<r\}.$

It does not include points at distance exactly $r$ from its center.

For example, with the usual distance on $\mathbb R$, the open ball $B_1(2)$ is the interval $(1,3)$; the endpoints are excluded because their distance from $2$ is exactly $1$.

6. Neighborhood

For $p\in X$ and $r>0$, the open ball $B_r(p)$ is a neighborhood of $p$. Its punctured neighborhood is

$B_r'(p)=B_r(p)\setminus\{p\}.$

7. Limit (accumulation) point

Let $E\subseteq X$. A point $p\in X$ is a limit point (or accumulation point) of $E$ if every punctured neighborhood of $p$ meets $E$; that is,

$\forall r>0,\quad B_r'(p)\cap E\ne\varnothing.$

The point $p$ need not itself belong to $E$.

8. Interior and exterior points

A point $p\in X$ is an interior point of $E\subseteq X$ if some open ball centered at $p$ lies entirely in $E$:

$\exists r>0\text{ such that }B_r(p)\subseteq E.$

A point is an exterior point of $E$ if it is an interior point of the complement $E^c=X\setminus E$. The interior of $E$, denoted $\operatorname{int}(E)$, is the set of its interior points. No point can be both interior to $E$ and interior to $E^c$.

9. Open and closed sets

A subset $E$ of a metric space $X$ is open if every point of $E$ is an interior point; equivalently,

$\forall x\in E,\ \exists r>0\text{ such that }B_r(x)\subseteq E.$

A set is closed if its complement in $X$ is open. Equivalently, $E$ is closed if it contains every limit point of $E$.

10. Covers

Let $E\subseteq X$, and let $\mathcal U=\{U_\alpha\}_{\alpha\in I}$ be a family of subsets of $X$.

1. $\mathcal U$ is a cover of $E$ if $E\subseteq\bigcup_{\alpha\in I}U_\alpha$.
2. It is an open cover if every $U_\alpha$ is open in $X$.
3. A subcover is a subfamily $\{U_\alpha\}_{\alpha\in J}$, where $J\subseteq I$, that still covers $E$.
4. It is a finite subcover if the index set $J$ is finite.

11. Compact sets

A subset $E$ of a metric space is compact if every open cover of $E$ has a finite subcover. In symbols, for every family $\mathcal U$ of open subsets of $X$ with $E\subseteq\bigcup_{U\in\mathcal U}U$, there is a finite subfamily $\mathcal V\subseteq\mathcal U$ such that

$E\subseteq\bigcup_{V\in\mathcal V}V.$

Order and completeness

12. Ordered sets

A (partially) ordered set is a set $A$ with a relation $\le$ that is reflexive, antisymmetric, and transitive:

1. Reflexivity: $a\le a$ for every $a\in A$.
2. Antisymmetry: if $a\le b$ and $b\le a$, then $a=b$.
3. Transitivity: if $a\le b$ and $b\le c$, then $a\le c$.

If any two elements are comparable, the order is a total order.

13. Bounds, supremum, and infimum

Let $(X,\le)$ be an ordered set and $A\subseteq X$. An element $u\in X$ is an upper bound of $A$ if $x\le u$ for every $x\in A$. The set $A$ is bounded above if it has an upper bound. A supremum (least upper bound) of $A$ is an upper bound $s$ such that $s\le u$ for every upper bound $u$ of $A$; it is denoted $\sup A$ when it exists.

A lower bound is defined by reversing the inequalities. A set with a lower bound is bounded below, and its infimum (greatest lower bound), denoted $\inf A$, is a lower bound that is greater than or equal to every lower bound of $A$, when it exists. A supremum or infimum need not belong to $A$.

14. Density of the rationals and irrationals

The real numbers are dense in themselves: between any two distinct real numbers there is another real number. More precisely, if $p,q\in\mathbb R$ and $p<q$, there is $r\in\mathbb R$ such that $p<r<q$.

Both the rational numbers $\mathbb Q$ and the irrational numbers $\mathbb R\setminus\mathbb Q$ are dense in $\mathbb R$: between any two distinct real numbers there is a rational number and also an irrational number.

15. Completeness of the real numbers

A sequence $(a_n)$ in a metric space $(X,d)$ is a Cauchy sequence if

$\forall\varepsilon>0,\ \exists N\in\mathbb N\text{ such that }m,n\ge N\implies d(a_m,a_n)<\varepsilon.$

A metric space is complete if every Cauchy sequence in it converges to a point of that space. The real numbers with their usual metric are complete; this is the completeness of $\mathbb R$.

Limits

16. Limit superior and limit inferior

For a sequence of sets $(A_n)$, its set-theoretic limit superior and limit inferior are

$$\begin{aligned} \limsup_{n\to\infty} A_n &= \bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty} A_k, \\ \liminf_{n\to\infty} A_n &= \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty} A_k. \end{aligned}$$

Thus, $x\in\limsup A_n$ exactly when $x$ belongs to infinitely many of the sets $A_n$, while $x\in\liminf A_n$ exactly when $x$ belongs to all sufficiently late sets.

For a real sequence $(a_n)$, the corresponding numerical definitions (with suprema and infima taken in the extended real numbers) are

$$\begin{aligned} \limsup_{n\to\infty} a_n &= \inf_{n\ge1}\sup_{k\ge n}a_k, \\ \liminf_{n\to\infty} a_n &= \sup_{n\ge1}\inf_{k\ge n}a_k. \end{aligned}$$

In brief

Algebraic operations, distance, order, and completeness are distinct pieces of language that work together in analysis; limsup and liminf summarize the long-term behavior of sequences.