The Timpson Lyceum · General Topology (Point Set Topology)

Closed Sets – Lecture 9 (Topology)

Closed Sets – Lecture 9

Overview

This lecture introduces closed sets, the natural companion to open sets, and defines them entirely in terms of set complements. A set is closed exactly when its complement is open — so "closed" is not the logical opposite of "open"; a set can be open, closed, both, or neither. After building intuition through examples in the standard topology on $\mathbb{R}$ and $\mathbb{R}^2$, we prove that closed balls are genuinely closed, review De Morgan's laws (with the logic behind them), and use those laws to show that closed sets satisfy three properties mirroring the axioms for open sets. The lecture closes with the Hausdorff separation property — the first true topological property used to classify spaces.

Key Concepts

Open set (review). A set is open precisely when it is a member of the topology $\tau$. The three topology axioms: $\varnothing$ and the whole space $X$ are open; any finite intersection of open sets is open; any arbitrary union of open sets is open.

Complement. For $U \subseteq X$, the complement is $U^{c} = X \setminus U$ — all points of $X$ not in $U$. We write the complement with a superscript $c$ (reserving the overbar notation for closure, introduced later).

Closed set. Let $(X,\tau)$ be a topological space and $A \subseteq X$. Then $A \text{ is closed} \iff A^{c} \text{ is open}.$ That is the entire definition — to test whether $A$ is closed, write down $A^{c}$ and ask whether it is open.

Open vs. closed are not opposites. A set may be:

• open only (e.g. an open interval $(a,b)$ in $\mathbb{R}$),
• closed only (e.g. a closed interval $[a,b]$),
• both (called clopen — e.g. $\varnothing$ and $X$, which are open and closed in every topology),
• neither (e.g. a half-open interval $[2,5)$, or $\mathbb{Z}\cup\{$random gaps$\}$).

Examples in the standard topology on $\mathbb{R}$ (open sets = unions of open intervals):

• $[a,b]$ is closed: its complement $(-\infty,a)\cup(b,\infty)$ is a union of open intervals, hence open.
• $\{2\}$ is closed: its complement $(-\infty,2)\cup(2,\infty)$ is open.
• $[0,1]\cup(2,3)$ is neither: it is not open (it cannot be built from open intervals because of the endpoints $0$ and $1$), and its complement contains the terminating point $2$, so the complement is not open either.
• $\mathbb{Z}$ is closed: its complement is the union of all open intervals $(n,n+1)$ between consecutive integers.
• $\mathbb{N}$ is closed for the same reason.
• $\mathbb{Q}$ is neither open nor closed: every open interval contains both rationals and irrationals, so neither $\mathbb{Q}$ nor its complement (the irrationals) can be open.

Open and closed balls. In a metric setting, the open ball of radius $r$ about $x$ is $B(x,r) = \{\, y : d(x,y) < r \,\},$ and the closed ball is $\overline{B}(x,r) = \{\, y : d(x,y) \le r \,\}.$ The only difference is whether the boundary ($d(x,y)=r$) is included. Open balls are open sets; closed balls are closed sets. In the standard topology on $\mathbb{R}^2$, open sets are unions of open balls (open disks); in $\mathbb{R}^3$ they are unions of open 3-balls, and so on.

De Morgan's laws (logic). Distributing a negation over a connective flips it: $\lnot(\alpha \land \beta) \equiv (\lnot\alpha)\lor(\lnot\beta), \qquad \lnot(\alpha \lor \beta) \equiv (\lnot\alpha)\land(\lnot\beta).$

De Morgan's laws (sets). Because $x\in A\cap B$ means $x\in A \land x\in B$, the same flip carries over to complements, for any number of sets: $\left(\bigcap_{i} A_i\right)^{c} = \bigcup_{i} A_i^{c}, \qquad \left(\bigcup_{i} A_i\right)^{c} = \bigcap_{i} A_i^{c}.$ Bringing the complement inside turns intersections into unions and unions into intersections.

Three properties of closed sets. For any topological space $(X,\tau)$:

• $\varnothing$ and $X$ are closed.
• Any finite union of closed sets is closed.
• Any arbitrary intersection of closed sets is closed.

These are exactly the open-set axioms with intersection and union swapped — and they follow immediately by taking complements and applying De Morgan's laws.

Hausdorff property. A space $(X,\tau)$ is Hausdorff if for any two distinct points $x \ne y$ there exist disjoint open sets (neighborhoods) $U \ni x$ and $V \ni y$ with $U \cap V = \varnothing$ — you can "house off" any two distinct points into separate neighborhoods. Key facts:

• The standard topology on $\mathbb{R}$ (and on $\mathbb{R}^n$) is Hausdorff, as is the lower-limit topology.
• In a Hausdorff space, every singleton $\{x\}$ is closed.
• A finite space is Hausdorff only if it carries the discrete topology.
• The finite-complement topology on an infinite set is not Hausdorff.

Worked Example

Claim. In the standard topology on $\mathbb{R}^2$, every closed ball $\overline{B}(x,r)$ is a closed set.

Strategy. Let $U = \overline{B}(x,r)^{c}$. To show $\overline{B}(x,r)$ is closed we show $U$ is open, i.e. that around every point $y\in U$ we can fit an open ball $B(y,\varepsilon)$ lying entirely in $U$; then $U$ is the union of all such balls, hence open (the "union lemma").

Pick any $y \in U$. Since $y \notin \overline{B}(x,r)$, we have $d(x,y) > r$, so $d(x,y) - r > 0$. Choose any radius with $0 < \varepsilon < d(x,y) - r.$ It remains to show $B(y,\varepsilon) \cap \overline{B}(x,r) = \varnothing$. Suppose, for contradiction, some point $z$ lies in both balls. By the triangle inequality, substituting the two ball conditions $d(x,z)\le r$ and $d(z,y)<\varepsilon$, and then $\varepsilon < d(x,y)-r$:

This gives $d(x,y) < d(x,y)$, a contradiction. Hence no such $z$ exists, so $B(y,\varepsilon)\subseteq U$. Taking the union over all $y\in U$ shows $U$ is open, and therefore $\overline{B}(x,r)$ is closed. $\blacksquare$

Summary

• A set $A$ is closed iff its complement $A^{c}$ is open — closed is defined through complements, not as the negation of open.
• "Open" and "closed" are independent properties: a set can be open, closed, both (clopen), or neither; $\varnothing$ and $X$ are always clopen.
• In the standard topology on $\mathbb{R}$: closed intervals, $\{2\}$, $\mathbb{Z}$, and $\mathbb{N}$ are closed; $\mathbb{Q}$ is neither; half-open or finitely-listed sets are typically neither.
• De Morgan's laws convert the open-set axioms into the closed-set properties: $\varnothing$ and $X$ closed, finite unions of closed sets closed, arbitrary intersections of closed sets closed.
• A closed ball is a closed set, proved by fitting a small open ball around each exterior point using the triangle inequality.
• A space is Hausdorff when any two distinct points have disjoint neighborhoods; in such spaces singletons are closed. The standard topology on $\mathbb{R}^n$ is Hausdorff, while the finite-complement topology on an infinite set is not.