Open and Closed Sets, Interior, Closure

The next layer of structure over a metric is the collection of open sets, the topology. It distinguishes points interior to a set, points on its edge, and points approached from outside. Convergence, compactness, and connectedness are all stated through open sets, each built from a single primitive: the ball $B(x,\delta )=\{y:d(x,y)<\delta \}$.

Open and closed sets

The radius $\delta$ is allowed to depend on $x$: points deep inside $V$ can use a large ball, points near the edge need a small one. The requirement is only that some positive radius works at each point, so that from anywhere in $V$ one can move a little in any direction and stay in $V$.

Openness and closedness are not opposites. Most sets are neither, and two sets are both.

Examples on $\mathbb{R}$.

• $(0,\infty )$ is open. Given $x>0$, take $\delta =x$; then $B(x,\delta )=(0,2x)\subseteq (0,\infty )$.
• $[0,\infty )$ is closed. Its complement $(-\infty ,0)$ is open by the same argument with $\delta =-x$ for $x<0$.
• $[0,1)$ is neither. No ball about $0$ avoids negative numbers, so it is not open; and $1$ is a point of the complement every ball of which meets $[0,1)$, so the complement is not open, hence $[0,1)$ is not closed.
• $\varnothing$ and $\mathbb{R}$ are both open and closed (each is the other's complement, and both are open).

An open set can always be pictured as a union of balls, one around each of its points. The two extreme sets, $\varnothing$ and $X$, are open in every metric space.

The topology axioms

Open sets are closed under two operations, and the asymmetry between them, all unions but only finite intersections, is the defining feature of a topology.

For (T2), a point $x$ in the intersection has a ball $B(x,{\delta }_j)\subseteq V_j$ for each of the finitely many $j$; the smallest radius $\delta ={\min }_j{\delta }_j$ is positive and its ball sits in every $V_j$. For (T3), a point of the union lies in some single $V_{\lambda }$, whose ball already witnesses openness. The union argument never needs a minimum, so it works for infinitely many sets; the intersection argument needs ${\min }_j{\delta }_j>0$, which can fail once there are infinitely many terms.

Complementation turns each axiom into its dual for closed sets, since $(\bigcup E_{\lambda }{)}^c=\bigcap E_{\lambda }^c$ and vice versa.

The names are justified by the fact that the open ball is open and the closed ball is closed, which still requires proof.

Take $y\in B(x,\delta )$ and set $\alpha =\delta -d(x,y)>0$. For any $z\in B(y,\alpha )$, the triangle inequality gives $d(x,z)\le d(x,y)+d(y,z)<d(x,y)+\alpha =\delta$, so $B(y,\alpha )\subseteq B(x,\delta )$. Every point of the ball thus has room around it, and the ball is open.

Interior, closure, and boundary

Three derived sets classify how a point relates to a set $A$: it can sit strictly inside, be approachable from within, or straddle the two sides.

The interior is open (it is a union of open balls) and is the largest open set inside $A$; the closure is closed (an intersection of closed sets) and is the smallest closed set containing $A$; and $A$ is open iff $A=A^{\circ }$, closed iff $A=\overline{A}$. The two ball-based characterizations below are what get used in practice.

The closure is thus everything approachable from $A$: every ball about $x$, however small, contains a point of $A$. The boundary consists of the points close to $A$ and to its complement at once. Equivalently $\partial A=\overline{A}\cap \overline{A^c}$.

The point $1\in A$ is on the boundary (every ball about it meets both $A$ and its complement) yet belongs to $A$; the point $0\notin A$ is on the boundary and does not. Boundary points may or may not lie in the set, and that is what makes $A$ neither open nor closed here.

Two more computations show the range of behavior.

• The rationals $\mathbb{Q}$ in $\mathbb{R}$. Every ball $(x-\delta ,x+\delta )$ contains both a rational and an irrational, so ${\mathbb{Q}}^{\circ }=\varnothing$, $\overline{\mathbb{Q}}=\mathbb{R}$, and $\partial \mathbb{Q}=\mathbb{R}$. A set can have empty interior and yet closure equal to the whole space; such a set is dense.
• An open disk in ${\mathbb{R}}^2$. For $A=B(0,1)$, the interior is $A$ itself (it is open), the closure is the closed disk $C(0,1)$, and the boundary is the unit circle. Here the boundary is a genuine edge, disjoint from the open set it bounds.

The last row shows a set whose closure adds a single limit point.

Limit points and isolated points

The closure adds to $A$ precisely the points approachable from it. Two terms distinguish points approached by other points of $A$ from points that are not.

Writing $A^{\prime }$ for the set of limit points, the closure decomposes as $\overline{A}=A\cup A^{\prime }$. A set is closed exactly when it contains all its limit points. Isolated points are the opposite extreme: under the discrete metric every point is isolated, since $B(x,\frac{1}{2})=\{x\}$, so $A^{\prime }=\varnothing$ and $\overline{A}=A$ for every $A$; every set is closed.

The subspace topology

Whether a set is open depends on the ambient space, because balls are computed there. Openness in a subspace reduces to openness in the whole space as follows.

The subspace balls are the ambient balls cut down to $Y$, $B_Y(x,\delta )=B_X(x,\delta )\cap Y$, and intersecting an open $V\subseteq X$ with $Y$ carries this through.

One special case: if $V\subseteq X$ is itself open, then a subset $U\subseteq V$ is open in $V$ iff it is open in $X$.¹