Interior, Closure, and Boundary

Intuition: Think of a painted region $A$ on a page.

• $\text{int}(A)$ is the part of $A$ that is "safely inside" — every point has a little room around it also in $A$. This is a union of open sets, so it is itself open. It is in fact the largest open set contained in $A$.
• $\overline{A}$ is $A$ together with all the points "just outside" that $A$ nearly touches. This is an intersection of closed sets, so it is itself closed. It is the smallest closed set containing $A$.
• $\partial A$ is the "edge": points that are in the closure but not in the interior — those touching both $A$ and its complement.

Always, $\text{int}(A) \subseteq A \subseteq \overline{A}$.

$\text{int}(A)$ is automatically open (as a union of open sets), and $\overline{A}$ is automatically closed (as an intersection of closed sets). The boundary $\partial A = \overline{A} \setminus \text{int}(A) = \overline{A} \cap \text{int}(A)^c$ is an intersection of two closed sets, hence closed.

Useful complement identities. For any subset $B \subseteq X$, the following identities hold and are used constantly: $X \setminus \text{int}(B) = \overline{X \setminus B}, \qquad X \setminus \overline{B} = \text{int}(X \setminus B).$ In words: the complement of the interior is the closure of the complement, and the complement of the closure is the interior of the complement. Both follow directly from the definitions of interior and closure as unions/intersections of open/closed sets, together with de Morgan's laws.

Intuition: A set equals its own interior exactly when it is open; a set equals its own closure exactly when it is closed. This gives operational tests: to check whether $A$ is open, compute $\text{int}(A)$ and compare with $A$; to check whether $A$ is closed, compute $\overline{A}$.

Intuition:

• $x \in \text{int}(A)$ means "some whole neighbourhood of $x$ lies inside $A$."
• $x \in \overline{A}$ means "every neighbourhood of $x$ hits $A$."

In a metric space, "neighbourhood" can be replaced by "open ball of some radius," so:

• $x \in \text{int}(A) \iff \exists \rho > 0 : B(x, \rho) \subseteq A$.
• $x \in \overline{A} \iff \forall \rho > 0 : A \cap B(x, \rho) \ne \emptyset$.

Intuition: $x_0$ is an accumulation point of $A$ if every neighbourhood of $x_0$ contains some other point of $A$ (not just $x_0$ itself). So $A$ "piles up" near $x_0$. The distinction from "$x_0 \in \overline{A}$" is subtle but important: $x_0 \in \overline{A}$ only requires that every neighbourhood hits $A$ somewhere, possibly only at $x_0$; $x_0 \in A'$ requires that every neighbourhood hits $A$ at a point different from $x_0$. The notion is due to Georg Cantor (1872).

Intuition: $x$ is isolated in $A$ if $x$ sits in $A$ with a little buffer around it containing no other points of $A$. Isolated points of $A$ are in $A$ but not in $A'$. Conversely, a point of $A$ either is isolated in $A$ or is an accumulation point of $A$.

Intuition: "Closed" means "contains all its limit points." This is the sequential flavour of closedness we will make precise in metric spaces shortly.

Intuition: In metric spaces, "closed" really does mean "closed under taking limits of sequences." This is how many textbooks first introduce closedness in $\mathbb{R}^n$. Note that in a general topological space — one that need not be metric — this sequential characterization can fail; one needs the more abstract notions of nets or filters.

Intuition: $S$ is dense in $X$ if every point of $X$ either lies in $S$ or is a limit of points from $S$. In a metric space, equivalently: every open ball in $X$ contains a point of $S$. The prototypical example is $\mathbb{Q} \subseteq \mathbb{R}$: the rationals are dense in the reals because every real number can be approximated to arbitrary precision by rationals.

Why "$\bigcap_{F \supseteq S, F \in \mathcal{F}} (F \cap A) = A$" is the right relative-density condition. When we restrict the topology on $X$ to $A$ via the subspace topology $\mathcal{T}_A$, the closed sets according to $\mathcal{T}_A$ are exactly the sets of the form $F \cap A$ with $F$ closed in $X$. Denoting by $\mathcal{F}_A$ the family of closed sets of $\mathcal{T}_A$, we have $\bigcap_{\substack{F \in \mathcal{F} \\ S \subseteq F}} (F \cap A) = \bigcap_{\substack{F \in \mathcal{F} \\ S \subseteq F \cap A}} (F \cap A) = \bigcap_{\substack{G \in \mathcal{F}_A \\ S \subseteq G}} G,$ which is the closure of $S$ computed in the subspace topology $\mathcal{T}_A$. So "$S$ dense in $A$" is really just "$S$ is dense in $A$ with respect to its own (inherited) topology."

Intuition: Being nowhere dense is a strong form of "smallness." It's not enough for $S$ itself to miss open balls; even the closure of $S$ — the fattest version of $S$ — must contain no open ball. A nowhere dense set has no "fat" parts anywhere.

Intuition: Meagre sets are "topologically small" — a countable bag of nowhere-dense sets. Comeagre sets are the topologically big ones — their complements are topologically small. In a "good" space (specifically, a complete metric space), these two classes do not overlap: a set cannot be both small and big at once. This is the content of the Baire Category Theorem, which we state at the end of this chapter.

The terminology "first category" / "second category" was introduced by René Baire in 1899. Note: "residual" has nothing to do with category theory.

Equivalent description of comeagre sets. A set $S \subseteq X$ is comeagre if and only if $S$ can be written as a countable intersection of open dense sets. Indeed: $S$ comeagre means $X \setminus S = \bigcup_{n \ge 1} A_n$ with each $A_n$ nowhere dense, which by (ND2) means $A_n^c$ contains an open dense set $S_n$; taking complements gives $S \supseteq \bigcap_{n \ge 1} S_n$. The reverse direction is similar. This reformulation is the version of "comeagre" used in the proof of the Baire Category Theorem.

Intuition: The letters stand for German: $G$ for Gebiet (region) and $\delta$ for Durchschnitt (intersection); $F$ for fermé (closed, French) and $\sigma$ for Summe (sum/union). So $G_\delta$ is "open with intersections" and $F_\sigma$ is "closed with unions."

These classes sit between open and closed: every open set is trivially $G_\delta$ (a single-term intersection), every closed set is trivially $F_\sigma$. But the classes are strictly larger: $\{0\} = \bigcap_{k=1}^\infty (-\tfrac{1}{k}, \tfrac{1}{k})$ is $G_\delta$ but (as a closed set in $\mathbb{R}$) it is certainly not open. Similarly, by complementation, $\mathbb{Q} = \bigcup_{q \in \mathbb{Q}} \{q\}$ is $F_\sigma$ but not closed.

These classes are important in analysis and measure theory because many "natural" sets (the set of continuity points of a function, say) turn out to be $G_\delta$ or $F_\sigma$ even when they are neither open nor closed.

Significance. A classical result (which we will not prove here) states that the set of points at which a function $f : \mathbb{R} \to \mathbb{R}$ is continuous is always a $G_\delta$ set. Conversely, there is no function on $\mathbb{R}$ whose set of continuity points is exactly $\mathbb{Q}$, because $\mathbb{Q}$ is not a $G_\delta$ set in $\mathbb{R}$ (the proof of this last fact uses the Baire Category Theorem).

Intuition: In a complete metric space, "topologically small" (meagre) sets really are small: they cannot exhaust the space, and their complements are dense. The Baire Category Theorem has extraordinary reach — it gives a non-constructive proof that the irrationals are uncountable (and nonmeagre), lies at the heart of functional analysis (Open Mapping, Closed Graph, and Uniform Boundedness theorems), and connects to the deep fact that "most" continuous functions are nowhere differentiable.

Game-theoretic interpretation (Choquet's game). A beautiful reformulation of the Baire property is in terms of a two-player game due to Gustave Choquet (1969). Let $(X, \mathcal{T})$ be a topological space. Two players, Alice and Bob, play as follows. Alice moves first, choosing a nonempty open set $U_0 \subseteq X$. Then Bob chooses a nonempty open $V_0 \subseteq U_0$; then Alice chooses nonempty open $U_1 \subseteq V_0$; and so on. The nested chain $X \supseteq U_0 \supseteq V_0 \supseteq U_1 \supseteq V_1 \supseteq U_2 \supseteq \cdots$ is produced. Alice wins if $\bigcap_{j \ge 0} U_j = \emptyset$; Bob wins if $\bigcap_{j \ge 0} V_j \ne \emptyset$ (note $\bigcap U_j = \bigcap V_j$).

In a nonempty complete metric space, Bob has a winning strategy (he can always shrink his ball by half and keep closure inside the previous ball, producing a Cauchy sequence that must converge by completeness). Conversely, if the space contains a nonempty meagre open set, Alice has a winning strategy. This game-theoretic dichotomy recovers the Baire Category Theorem as a corollary. John Oxtoby proved the further equivalence: every comeagre set is dense if and only if Alice has no winning strategy.

Payoff: the irrationals are nonmeagre. Since $\mathbb{Q}$ is meagre in $\mathbb{R}$ and $\mathbb{R}$ itself is nonmeagre (Baire), we can deduce that $\mathbb{R} \setminus \mathbb{Q}$ must be nonmeagre: if it were meagre, $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$ would be a union of two meagre sets, hence meagre — contradicting Baire. This also shows that there is no way to write $\mathbb{R} \setminus \mathbb{Q}$ as a countable union of closed sets, i.e., the irrationals are not $F_\sigma$.