Open and Closed Sets in Metric Spaces

Understanding open and closed sets is not merely an exercise in definition-memorization; it is the essential first step toward mastering analysis and topology. These concepts form the very language used to describe convergence, continuity, and compactness, which are the cornerstones of modern mathematical analysis. By characterizing sets through the behavior of points within and around them, you gain a powerful framework for dissecting the structure of abstract spaces.

Foundations: Metric Spaces and Neighborhoods

The stage for our discussion is a metric space. A metric space is a set $X$ equipped with a metric $d:X\times X\to \mathbb{R}$, which is a function satisfying three key properties for all points $x,y,z\in X$: non-negativity ($d(x,y)\ge 0$), identity of indiscernibles ($d(x,y)=0$ if and only if $x=y$), symmetry ($d(x,y)=d(y,x)$), and the triangle inequality ($d(x,z)\le d(x,y)+d(y,z)$). Common examples include the real line $\mathbb{R}$ with $d(x,y)=|x-y|$, and the plane ${\mathbb{R}}^2$ with the Euclidean distance.

The core tool for probing a metric space is the concept of a neighborhood. Given a point $p\in X$ and a radius $r>0$, the open ball centered at $p$ is defined as $B_r(p)=\{x\in X:d(x,p)<r\}$. Think of this as all points within a distance $r$ of $p$, excluding the boundary. An open ball is the fundamental "neighborhood" of a point. A set $N$ is called a neighborhood of $p$ if it contains some open ball centered at $p$. This idea allows us to talk about points that are "close to" or "surround" $p$.

Characterizing Open Sets via Interior Points

We can now define the most important type of set. A subset $U$ of a metric space $X$ is open if every point of $U$ is an interior point. A point $p\in U$ is an interior point of $U$ if there exists some neighborhood of $p$ that is entirely contained within $U$. In formal terms, there exists an $r>0$ such that $B_r(p)\subseteq U$.

This definition captures the intuitive idea that an open set does not include its "edge." For any point you pick in an open set, you can move a little in any direction and still remain inside the set. Classic examples include:

• Any open interval $(a,b)$ in $\mathbb{R}$.
• The entire space $X$ and the empty set $\varnothing$ are always open.
• Any union of open sets is open.
• The finite intersection of open sets is open.

The collection of all open subsets of a metric space is called its topology, and it defines the space's continuous structure.

Characterizing Closed Sets via Limit Points

The dual notion is that of a closed set. The most intuitive definition—a set that contains all its boundary points—is useful but imprecise in this context. A more powerful characterization uses the idea of limit points (or accumulation points).

A point $p\in X$ is a limit point of a subset $A\subseteq X$ if every neighborhood of $p$ contains at least one point of $A$ different from $p$ itself. Symbolically, for every $r>0$, we have $(B_r(p)\setminus \{p\})\cap A\ne \varnothing$. A limit point of $A$ may or may not be in $A$. A set $C\subseteq X$ is defined to be closed if it contains all of its limit points.

Consider the set $A=\{1/n:n\in \mathbb{N}\}$ in $\mathbb{R}$. The point $0$ is a limit point of $A$ because any interval around $0$ will contain infinitely many points of the form $1/n$. Since $0\notin A$, the set $A$ is not closed. If we form $C=A\cup \{0\}$, then $C$ contains all its limit points (the only one being $0$) and is therefore closed.

The Relationship and Key Operators

Open and closed sets are fundamentally related through complements. In any metric space $X$, a subset $C$ is closed if and only if its complement, $X\setminus C$, is open. This complementarity is a crucial theorem that connects the two concepts and simplifies many proofs.

Two natural operators arise from these definitions: the closure and the interior.

1. The closure of a set $A$, denoted $\overline{A}$, is the union of $A$ and the set of all its limit points. It is the smallest closed set containing $A$. You can think of it as $A$ plus its "fence."
2. The interior of $A$, denoted $A^{\circ }$, is the set of all interior points of $A$. It is the largest open set contained within $A$. Think of it as $A$ minus its "skin."

The boundary of $A$, denoted $\partial A$, is the set of points that are in the closure of $A$ but not in its interior: $\partial A=\overline{A}\setminus A^{\circ }$. A point is in the boundary if every neighborhood of it intersects both $A$ and its complement.

Dense Subsets and Approximation Theory

The concepts of closure and limit points lead directly to the idea of density, which is vital for approximation. A subset $D$ of a metric space $X$ is said to be dense in $X$ if every point in $X$ is either in $D$ or is a limit point of $D$. Equivalently, the closure of $D$ is the whole space: $\overline{D}=X$. A third, highly useful formulation is: $D$ is dense if and only if every non-empty open set in $X$ contains at least one point of $D$.

Density is the formal machinery behind approximation theory. For example, the rational numbers $\mathbb{Q}$ are dense in the real numbers $\mathbb{R}$. This means any irrational number, like $\pi$ or $\sqrt{2}$, can be approximated arbitrarily closely by a rational number. In functional analysis, the density of polynomials in the space of continuous functions (per the Weierstrass Approximation Theorem) guarantees that any continuous function on a closed interval can be uniformly approximated by a polynomial—a result with profound implications for numerical methods and signal processing.

Common Pitfalls

1. Confusing "Not Open" with "Closed": A set can be neither open nor closed. In $\mathbb{R}$, the interval $[0,1)$ is not open (because $0$ is not an interior point) and not closed (because $1$ is a limit point not contained in the set). "Open" and "closed" are not logical opposites; they are complements in the specific sense that a set is closed iff its complement is open. The true opposites are "open" and "not open."
2. Misidentifying Limit Points: A common error is to think a point must have infinitely many points of $A$ in every neighborhood to be a limit point. The definition only requires at least one point of $A$ (different from $p$) in every neighborhood. This logically implies there must be infinitely many, but it's important to reason from the definition. Also, an isolated point of $A$ (where you can find a neighborhood containing no other points of $A$) is not a limit point of $A$.
3. Assuming Properties from ${\mathbb{R}}^n$ Hold Generally: In ${\mathbb{R}}^n$, a set is often "closed and bounded," which implies compactness. In a general metric space, "closed and bounded" does not necessarily imply compactness. Always verify which theorems depend on the specific structure of ${\mathbb{R}}^n$ (like the Heine-Borel Theorem) and which are true in all metric spaces.
4. Misinterpreting Density: Density does not mean "almost all points." A dense set can be "small" in other senses, like the rationals being countable within the uncountable reals. It means the set is ubiquitously scattered throughout the space, ensuring you can find a member of the set arbitrarily close to any point you choose.

Summary

• Open sets are defined via interior points: a set $U$ is open if for every $p\in U$, there exists a neighborhood of $p$ entirely contained within $U$.
• Closed sets are defined via limit points: a set $C$ is closed if it contains all points $p$ such that every neighborhood of $p$ intersects $C$ (in a point other than $p$ itself).
• Open and closed sets are complementary: $C$ is closed if and only if $X\setminus C$ is open. This is a fundamental duality.
• The closure $\overline{A}$ of a set $A$ is the smallest closed set containing $A$ (formed by adding all limit points). The interior $A^{\circ }$ is the largest open set contained in $A$.
• A set $D$ is dense in $X$ if its closure is $X$, meaning every point in $X$ can be approximated arbitrarily closely by points from $D$. This concept bridges pure set theory with practical approximation theory.

Mastering these definitions and their interrelationships equips you with the precise vocabulary needed to navigate deeper results in analysis, topology, and functional analysis, where questions of convergence and approximation are paramount.