Countable Box Covers of Subsets of Rn

From Bounded Regions to Countable Covers

A box cover is a way to describe a set using simpler axis-parallel pieces. In this tutorial, the pieces are half-open boxes, and the quantity we calculate for each box is its coordinate volume. The goal is geometric: construct covers and add the coordinate volumes of their boxes. No measure or integral is being defined here, and a cover's total coordinate volume need not equal the size of the set it covers.

The half-open convention assigns a shared grid boundary to one cell rather than both adjacent cells. It also requires attention to the upper endpoint: \(b_i\) is not in \([a_i,b_i)\). We will choose the grid indices so that points at the edge of the bounded region are included in a cell to their right.

Finite Grid Covers for Bounded Sets

Every bounded set is contained in some cube \([-N,N]^n\), with \(N\) a positive integer. Divide each coordinate axis into intervals of length \(1/q\), where \(q\) is a positive integer. The cells that meet \([-N,N]^n\) can be listed using integer indices from \(-Nq\) to \(Nq\), inclusive. That final index is needed to cover the point whose coordinate equals \(N\).

Proof. Fix \(x=(x_1,\ldots,x_n)\in[-N,N]^n\). For each coordinate \(i\), the integer \(k_i=\lfloor qx_i\rfloor\) satisfies \(k_i\le qx_i<k_i+1\). Since \(-Nq\le qx_i\le Nq\), we have \(-Nq\le k_i\le Nq\); in particular, if \(x_i=N\), then \(k_i=Nq\). Thus \(x_i\in[k_i/q,(k_i+1)/q)\) for every \(i\), and \(x\in B_k\). There are \(2Nq+1\) choices for each of the \(n\) indices, giving \((2Nq+1)^n\) boxes. Each box has coordinate volume \((1/q)^n\), so the total is \((2Nq+1)^n/q^n=(2N+1/q)^n\). \(\square\)

This bound is an estimate from a particular finite cover; it is not a claim that the covered set has that coordinate volume. Increasing \(q\) makes the grid finer, but the total for this cover approaches \((2N)^n\), rather than zero. A bounded set may be large, so boundedness alone does not provide small-total-volume covers.

Countable Sets: Arbitrarily Small Total Coordinate Volume

For a countable set, a different construction is available. Give the \(j\)-th point a box whose coordinate volume is a prescribed part of a convergent series. To make a cube in \(\mathbb{R}^n\) have volume \(c\), choose its side length to be \(c^{1/n}\). Centering such a cube at the point and making it half-open ensures that the point itself is included.

Proof. If \(E\) is empty, the empty collection covers it and has total coordinate volume zero. Otherwise list its distinct points as \(x_j=(x_{j,1},\ldots,x_{j,n})\), with \(j=1,\ldots,m\) if \(E\) is finite, or \(j\ge1\) if it is infinite. Set \(s_j=(\varepsilon 2^{-j-1})^{1/n}\), and define \(B_j=\prod_{i=1}^n[x_{j,i}-s_j/2,x_{j,i}+s_j/2)\). Each coordinate interval has positive length \(s_j\), and \(x_{j,i}\) lies in that interval because its displacement from the left endpoint is \(s_j/2\), which is nonnegative and strictly less than \(s_j\). Hence \(x_j\in B_j\), so the boxes cover \(E\). Their total coordinate volume is \(\sum_j s_j^n=\sum_j\varepsilon 2^{-j-1}\). For an infinite list this sum is \(\varepsilon/2\), since \(\sum_{j=1}^{\infty}2^{-j-1}=1/2\). For a finite list its sum is no greater than \(\varepsilon/2\), because it is a partial sum of nonnegative terms. Finally, \(\varepsilon/2<\varepsilon\). \(\square\)

A Countable Shell Cover for Any Unbounded Set

The countable-point construction does not apply to every unbounded set: an unbounded set need not be countable. A general construction uses expanding cubes. Write \(C_j=[-j,j]^n\) for positive integers \(j\). Every point of \(\mathbb{R}^n\) belongs to some \(C_j\), so these cubes provide bounded stages that exhaust the space. For a set \(E\), the pieces \(E\cap C_j\) form a countable cover of \(E\); each can be enclosed in a single half-open box.

Proof. Fix \(x\in E\). Since its finitely many coordinates are real, there is a positive integer \(j\) with \(|x_i|\le j\) for all \(i\). Thus \(x\in C_j\). For each coordinate, \(-j\le x_i\le j<j+1\), so \(x_i\in[-j,j+1)\) and \(x\in Q_j\). This proves coverage. Each \(Q_j\) has side length \((j+1)-(-j)=2j+1\) in every coordinate, and therefore \(v(Q_j)=(2j+1)^n\). Since \(n\ge1\), each term is at least \(1\), so the partial sum through \(j=M\) is at least \(M\). These partial sums are unbounded as \(M\to\infty\), proving that the total is infinite. \(\square\)

A Common Mistake: The Side Length Is Not the Volume

To allocate a target coordinate volume \(c_j\) to a cube in \(\mathbb{R}^n\), one must set its side length to \(c_j^{1/n}\), not \(c_j\). A side length \(c_j\) gives volume \(c_j^n\). In dimensions greater than one, confusing these quantities changes the assigned volumes and may invalidate the intended estimate.

There is a separate endpoint pitfall: an open cube centered at a point is not a half-open box under the definition used here. The correction is to use \(\prod_{i=1}^n[x_{j,i}-s_j/2,x_{j,i}+s_j/2)\). Its left endpoints are included, its right endpoints are excluded, and the center lies strictly between them in each coordinate. This verifies both the required box form and coverage.

The constructions serve different purposes. A grid gives a finite, explicit cover of any bounded set inside a known cube. The countable-point construction gives arbitrarily small total coordinate volume for countable sets, whether bounded or unbounded. Expanding boxes give a countable cover of every subset of \(\mathbb{R}^n\), but the particular shell cover has infinite total coordinate volume. The amount calculated is always the sum for the chosen cover; estimating or defining an outer measure from such covers is a separate step.