The Standard Topology of R^n

Topology on $\mathbb{R}^n$ is where abstract definitions become concrete intuition. The axioms of §4.1 are realized on Euclidean space by one simple object: the open ball. From open balls, you build open sets; from open sets, you get interior, closure, boundary; and from those, you reach compactness and connectedness, the two topological properties that drive virtually every existence proof in real analysis. The Heine-Borel theorem, the highlight of this section, gives a remarkably clean characterization: in $\mathbb{R}^n$, "compact" means "closed and bounded."

Open sets via open balls

The open ball of radius $r>0$ centred at $\mathbf{a}\in\mathbb{R}^n$ is B(\mathbf{a},r)=\{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}-\mathbf{a}\|0 with $B(\mathbf{x},r)\subseteq U$. This recovers the intuitive notion: a set is open if you can wiggle around every point a little without leaving the set. It is straightforward to check the three topology axioms hold for the collection of all such open sets, this is the standard topology on $\mathbb{R}^n$.

Interior, closure, boundary

For any $A\subseteq\mathbb{R}^n$:

• The interior $\operatorname{int}(A)$ is the largest open subset of $A$, equivalently, the set of points in $A$ that have an entire open ball inside $A$.
• The closure $\overline{A}$ is the smallest closed set containing $A$, equivalently, $A$ together with all its limit points.
• The boundary $\partial A=\overline{A}\setminus\operatorname{int}(A)$ is the points that are "right on the edge", in the closure but not the interior.

For A=(0,1]\subseteq\mathbb{R}: $\operatorname{int}(A)=(0,1)$, \overline{A}=[0,1], $\partial A=\{0,1\}$. The interior excludes the endpoint $1$ (no full open ball around $1$ stays in $A$), the closure adds the missing endpoint $0$, and the boundary collects both endpoints.

The Heine-Borel theorem

A subset $K\subseteq\mathbb{R}^n$ is compact when every open cover has a finite subcover. The Heine-Borel theorem says: a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded. This characterization is special to finite-dimensional Euclidean space, it fails dramatically in infinite-dimensional spaces, where closed-bounded sets need not be compact.

Compactness is the topological cousin of finiteness: continuous functions on a compact set achieve their max and min (Extreme Value Theorem); sequences in compact sets always have convergent subsequences (Bolzano-Weierstrass); compactness propagates under continuous images. Almost every existence theorem in analysis, existence of solutions to ODEs, existence of optimal points in optimization, existence of equilibria in game theory, reduces to a compactness argument somewhere.

The set-Venn widget above lets you visualise unions, intersections, and complements of regions in the plane. In $\mathbb{R}^2$, try to picture the interior, closure, and boundary of a closed disk minus an open disk inside (an annulus). The boundary consists of two circles; the interior is the open annulus; the closure adds both boundary circles back in.

• Optimization, Extreme Value Theorem: Every constrained optimization problem with a continuous objective on a closed-bounded feasible region has a maximum and a minimum. Linear programming, convex optimization, and global-optimization solvers all rely on this guarantee. When the feasible region is unbounded or open, an optimum may not exist, producing the famous "infeasible / unbounded" error states.
• Game theory, Nash equilibrium existence: Nash's 1950 theorem uses a fixed-point argument on a compact (closed-bounded) simplex of mixed strategies. Without Heine-Borel and the resulting fixed-point theorem (Brouwer), the entire field of non-cooperative game theory loses its foundational existence result.
• ODE theory, Peano existence theorem: The existence theorem for solutions of $x'=f(t,x)$ with continuous $f$ uses Arzelà-Ascoli, itself a compactness statement about families of equicontinuous functions on a compact interval. Every initial-value-problem solver in numerical software ultimately rests on this compactness.

Pause and think: Why is the open interval $(0,1)$ in $\mathbb{R}$ NOT compact? Try the open cover $\{(1/n, 1):n=2,3,\ldots\}$. Every point of $(0,1)$ is in some $(1/n,1)$, so it is a cover. Can you extract a finite subcover? No, any finite collection has a smallest lower endpoint $1/N$, and points in $(0, 1/N)$ are uncovered. So $(0,1)$ admits an open cover with no finite subcover, not compact, consistent with Heine-Borel (it is bounded but NOT closed).

Try it

• Predict first: is the integer lattice $\mathbb{Z}\subseteq\mathbb{R}$ closed? open? compact? (Answer: closed but not compact, since unbounded. Also not open: no open ball around an integer stays in $\mathbb{Z}$.)
• Find $\operatorname{int}([0,1)\cup\{2\})$ in $\mathbb{R}$. (Answer: $(0,1)$. The point 2 is isolated, so no open ball around it stays in the set.)
• Predict: is the union of two compact subsets of $\mathbb{R}^n$ compact? (Yes; the union is still closed and bounded.) Is the intersection of two compact sets compact? (Yes; closed under intersections and contained in a bounded set.) Is the union of countably many compact sets compact? (Not necessarily; \bigcup_n[0,n]=[0,\infty) is unbounded.)
• Show: [0,1]\times[0,1] is compact in $\mathbb{R}^2$. (Closed: it contains its boundary. Bounded: contained in $B(\mathbf{0},2)$. By Heine-Borel, compact.)
• Trap: in $\mathbb{Q}$ (rationals with subspace topology from $\mathbb{R}$), the set $\{q\in\mathbb{Q}:q^2<2\}$ is closed and bounded but NOT compact. (No finite subcover of $\{(-\sqrt{2}+1/n,\sqrt{2}-1/n)\cap\mathbb{Q}:n\geq 1\}$.) Heine-Borel is special to $\mathbb{R}^n$, not to general metric spaces.

A trap to watch for

Heine-Borel is a theorem about $\mathbb{R}^n$, not about all metric spaces. In an infinite-dimensional Banach space (e.g. $\ell^2$, square-summable sequences), the closed unit ball is NOT compact, this is Riesz's theorem. In $\mathbb{Q}$ as a subspace of $\mathbb{R}$, closed-bounded sets need not be compact (the sequence $1, 1.4, 1.41, 1.414, \ldots$ has no convergent subsequence in $\mathbb{Q}$). The takeaway: use "closed + bounded" for compactness only in finite-dimensional Euclidean space.

What you now know

You can identify open and closed subsets of $\mathbb{R}^n$, compute interior/closure/boundary, and apply Heine-Borel to test compactness. The next section generalizes: metric spaces are the natural setting where "distance" still makes sense (so open balls still work) but where Heine-Borel might fail.

Mark section complete →

References

• Garrity, T. (2002). All the Mathematics You Missed. Cambridge UP, ch. 4.
• Munkres, J. R. (2000). Topology (2nd ed.). Prentice-Hall, ch. 3.
• Willard, S. (1970). General Topology. Addison-Wesley, ch. 3.
• Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 2.
• Kelley, J. L. (1955). General Topology. Van Nostrand, ch. 5.