Problem 5 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

1-17. Construct a set \(A \subset[0,1] \times[0,1]\) such that \(A\) contains at most one point on each horizontal and each vertical line but boundary \(A=[0,1] \times[0,1] .\) Hint: It suffices to ensure that \(A\) contains points in each quarter of the square \([0,1] \times[0,1]\) and also in each sixteenth, etc.

Step-by-Step Solution

Answer

Divide the square repeatedly and place points ensuring each vertical and horizontal line contains at most one point.

Step 1: Understanding the Problem

The goal is to construct a set, denoted as \(A\), within the unit square \([0,1] \times [0,1]\). This set should contain at most one point on each horizontal and each vertical line while still including points across all divisions of the square.

Step 2: Divide the Square

First, divide the unit square \([0,1] \times [0,1]\) into four equal quarters, each of dimension \([0, 0.5] \times [0, 0.5]\). Ensure that the set \(A\) has points in each of these quarters.

Step 3: Ensure Points in Subdivisions

Next, divide each quarter into four smaller squares of dimension \([0, 0.25] \times [0, 0.25]\). Continue ensuring that \(A\) has points in each of these subdivisions. Repeat for 16th divisions, etc.

Step 4: Placement Strategy

Choose points systematically to place in each subdivision. For example, place a point in each quarter of the initial square, then place points in subdivisions ensuring each vertical and horizontal line contains at most one point.

Step 5: Points Distribution Check

Verify that as the subdivisions get smaller, each uses a unique horizontal and vertical placement. This prevents having more than one point align when scaling down infinitely.

Key Concepts

set theory

The foundation of this exercise lies in **set theory**. Set theory, at its core, delineates the study of collections of objects, which are termed as 'sets'.
A set is a collection of distinct elements or members. Here, the set in question, denoted as \(A\), is a subset of the Cartesian product \([0,1] \times [0,1]\). This means \(A\) is a collection of points inside the unit square.
Each point can be represented as \((a, b)\), where \(a\) and \(b\) fall between 0 and 1.
Importantly, the task is to construct \(A\) such that:

• \(A\) contains at most one point on each horizontal and vertical line within the unit square.
• Points need to be distributed in a manner that scales down to smaller fractions of the square.
  
  Understanding this approach is crucial in set theory because it leverages the concept of infinity and divisions.

By dividing and ensuring a point exists in smaller subsets recursively, we ensure the comprehensiveness of set \(A\) while adhering to the conditions.

geometry of unit square

Delving into the **geometry of the unit square** enriches our understanding of the problem.
The unit square \([0,1] \times [0,1]\) spans a plane from \((0,0)\) to \((1,1)\).

To comprehensively distribute points within this square, we must first subdivide it into smaller sections:

• Initial Division: Divide the unit square into four equal quarters: \([0, 0.5] \times [0, 0.5]\), \([0.5, 1] \times [0, 0.5]\), \([0, 0.5] \times [0.5, 1]\), and \([0.5, 1] \times [0.5, 1]\).
• Secondary Division: Each of these quarters is further divided into four parts, creating 16 smaller squares.
  
  This geometric breakdown is kaleidoscopic in nature.

By continually halving each subdivision, we enable space for point placement that respects the unique alignment criteria.

• In layman's terms, it likens the shrinking box technique, integrating sets and geometry harmoniously.

point distribution

Critical to the solution is **point distribution** within these geometric subdivisions.
Points should be strategically placed to satisfy both the alignment and coverage requirements.

Let's see how to achieve such placement:

• Start by placing one point in each quarter of the initial division while adhering to the 'one point per line' rule.
• Repeat the process for the subsequent subdivisions of each quarter.

This iterative strategy means:

• Every smaller square learns from the larger division's placements.
• As subdivisions shrink, we ensure no two points share the same horizontal or vertical alignment, spread uniformly without crowding a specific line.
• This meticulous process amalgamates set theory and geometry, providing an infinity practical approach proven by ensuring unique situations even as we dive infinitely smaller.
  
  Remember, the key lies in systematic and recursive distribution.

This emphasizes a comprehensive understanding of set division, geometrical spread and strategic point placement.