Real Analysis: Metric Spaces and Topology

Real analysis does not stop at the real line. Many of its most powerful ideas were shaped to work in abstract settings where “distance” is defined in a flexible way and where familiar geometric intuition must be replaced with precise definitions. Metric spaces and topology provide that foundation. They clarify what it means to converge, to be continuous, and to have structure like compactness or connectedness, all without relying on coordinates.

At the center of this viewpoint is a simple shift: instead of studying functions on $\mathbb{R}$ because we can draw them, we study functions on spaces where the essential analytic notions still make sense.

Metric spaces: the language of distance

A metric space is a set $X$ equipped with a distance function (a metric) $d:X\times X\to \mathbb{R}$ satisfying:

1. $d(x,y)\ge 0$, and $d(x,y)=0$ iff $x=y$
2. $d(x,y)=d(y,x)$
3. Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$

This small list drives much of modern analysis. Once you can measure distance, you can define neighborhoods, limits, continuity, and more.

Examples that shape intuition

• Euclidean space $({\mathbb{R}}^n,d_2)$ with $d_2(x,y)=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$.
• The sup metric on ${\mathbb{R}}^n$: $d_{\infty }(x,y)={\max }_i|x_i-y_i|$, useful for uniform estimates.
• Function spaces: on continuous functions $C([a,b])$, a common metric is $d(f,g)={\sup }_{x\in [a,b]}|f(x)-g(x)|$, which encodes uniform convergence.

These different metrics may yield different notions of “closeness,” and therefore different convergence and continuity behavior. Understanding the metric is understanding the analysis.

Open and closed sets: topology from distance

Topology formalizes the idea of nearness without tying it to algebraic structure. In a metric space, the key building blocks are open balls: \[ B(x,r)=\{y\in X : d(x,y)<r\}. \]

A set $U\subseteq X$ is open if for every $x\in U$ there exists $r>0$ such that $B(x,r)\subseteq U$. A set is closed if its complement is open. Equivalently in metric spaces, a set is closed if it contains all its limit points.

These definitions are not cosmetic. They determine:

• which sequences can “stay inside” a set and still converge to a point outside it,
• what it means for a function to be continuous,
• how properties like compactness behave.

Interior, closure, and boundary

Given $A\subseteq X$:

• The interior $A^{\circ }$ is the largest open set contained in $A$.
• The closure $\overline{A}$ is the smallest closed set containing $A$.
• The boundary $\partial A=\overline{A}\setminus A^{\circ }$ captures points that cannot be isolated from either $A$ or its complement.

In analysis, closures matter because limits often land on the boundary. For example, if a sequence in $A$ converges, its limit must lie in $\overline{A}$.

Convergence and Cauchy sequences

A sequence $(x_n)$ converges to $x$ if $d(x_n,x)\to 0$. This definition generalizes the familiar $ϵ$-definition on $\mathbb{R}$ and immediately connects with open sets: $x_n\to x$ precisely when for every open neighborhood $U$ of $x$, eventually $x_n\in U$.

A sequence is Cauchy if for every $ϵ>0$ there exists $N$ such that for all $m,n\ge N$, $d(x_m,x_n)<ϵ$. Cauchy sequences are those whose terms eventually become mutually close, even if you do not yet know what point they converge to.

This distinction matters because, in some spaces, Cauchy sequences need not converge.

Completeness: when Cauchy sequences converge

A metric space $(X,d)$ is complete if every Cauchy sequence converges to a point in $X$. Completeness is a central condition in real analysis because many existence theorems can be phrased as: “a Cauchy process has a limit.”

Why completeness is not optional

The rational numbers $\mathbb{Q}$ with the usual metric are not complete. There are Cauchy sequences of rationals that converge to irrational limits, which do not live in $\mathbb{Q}$. This is one reason $\mathbb{R}$ is built as a completion of $\mathbb{Q}$.

In functional analysis and differential equations, completeness is often baked into the setting via Banach spaces (complete normed spaces). Without completeness, fixed point arguments and limit constructions can fail because the “would-be limit” falls outside the space.

Compactness: finiteness hidden in infinity

Compactness is one of the most consequential topological properties in analysis. In metric spaces, a set $K$ is compact if every open cover of $K$ has a finite subcover. While this definition is abstract, metric spaces provide practical equivalents:

• Sequential compactness: every sequence in $K$ has a convergent subsequence with limit in $K$.
• Total boundedness plus completeness: $K$ is compact iff it is complete and totally bounded (can be covered by finitely many balls of radius $ϵ$ for every $ϵ>0$).

Compactness as a tool

Compactness converts infinite behavior into finite control. Some standard consequences in metric spaces:

• Continuous functions on compact sets attain extrema: a continuous $f:K\to \mathbb{R}$ reaches a maximum and a minimum.
• Uniform continuity: every continuous $f:K\to \mathbb{R}$ is uniformly continuous, meaning a single $\delta$ works for all points in $K$ for a given $ϵ$.

These facts are not merely “nice.” They are used to justify exchanging limits, proving existence of optimizers, and ensuring stability of estimates.

Connectedness: when a space cannot be split

A space (or subset) is connected if it cannot be written as the union of two nonempty disjoint open sets (in the subspace topology). Connectedness prevents a domain from splitting into separated pieces.

In $\mathbb{R}$, connected sets are exactly intervals. This characterization underlies many classic theorems, including the intermediate value property: if $f$ is continuous and the domain is connected, the image is connected, which in $\mathbb{R}$ means it is an interval. That is the topological heart of the Intermediate Value Theorem.

Connectedness matters whenever an argument depends on “no gaps,” such as existence of roots, path arguments, or ensuring that continuity forces values to sweep through intermediate levels.

Continuity via topology

In metric spaces, a function $f:X\to Y$ between metric spaces is continuous at $x$ if for every $ϵ>0$ there exists $\delta >0$ such that $d_X(x,y)<\delta$ implies $d_Y(f(x),f(y))<ϵ$. Topology reframes this:

• $f$ is continuous iff the preimage of every open set in $Y$ is open in $X$.
• Equivalently, $f$ is continuous iff it preserves limits of sequences: $x_n\to x$ implies $f(x_n)\to f(x)$ (in metric spaces).

This flexibility is practical. The $ϵ$-$\delta$ definition is ideal for estimates, while the open-set definition is ideal for structural reasoning.

How these ideas work together in real analysis

The strength of metric spaces and topology is not in isolated definitions but in how they interlock:

• Convergence depends on open sets (neighborhoods).
• Closed sets capture where limits can land.
• Completeness ensures Cauchy sequences do not “escape.”
• Compactness upgrades pointwise control to uniform control and guarantees extrema.
• Connectedness prevents a space from decomposing in ways that break intermediate value arguments.

When real analysis moves to abstract settings, the goal is often to recover the theorems you trust on $\mathbb{R}$ by identifying which properties actually powered those theorems. Metric spaces and topology provide the exact vocabulary for that translation.