Problem 18 - Basic Analysis: Introduction to Real Analysis

Question

For every \(x \in \mathbb{R}^{n}\) and every \(\delta>0\) define the "rectangle" \(R(x, \delta):=\left(x_{1}-\delta, x_{1}+\delta\right) \times\left(x_{2}-\right.\) \(\left.\delta, x_{2}+\delta\right) \times \cdots \times\left(x_{n}-\delta, x_{n}+\delta\right) .\) Show that these sets generate the same open sets as the balls in standard metric. That is, show that a set \(U \subset \mathbb{R}^{n}\) is open in the sense of the standard metric if and only if for every point \(x \in U,\) there exists \(a \delta>0\) such that \(R(x, \delta) \subset U\)

Step-by-Step Solution

Answer

Open sets in rectangles and balls in \(\mathbb{R}^n\) are equivalent definitions of openness.

Step 1: Understanding Open Sets in Standard Metric

An open set in the standard metric on \(\mathbb{R}^n\) is defined such that for every point \(x \in U\), there exists an \(\epsilon > 0\) such that the open ball \(B(x, \epsilon) = \{y \in \mathbb{R}^n : \| y - x \| < \epsilon \}\) is contained within \(U\). We aim to show that rectangles \(R(x, \delta)\) can serve the same purpose.

Step 2: Showing Rectangle Can Contain Ball

Consider a point \(x \in U\). There exists an \(\epsilon > 0\) such that \(B(x, \epsilon) \subset U\) because \(U\) is open. The rectangle \(R(x, \delta)\) can be chosen such that it fits inside \(B(x, \epsilon)\) by selecting \(\delta < \frac{\epsilon}{\sqrt{n}}\). This ensures that the rectangle, with its boundaries tightly hugging the ball, is contained within \(U\).

Step 3: Showing Ball Can Contain Rectangle

If \(R(x, \delta) \subset U\), then for a small enough positive \(\delta\), we can fit a ball inside \(R(x, \delta)\). Specifically, choose \(\epsilon = \delta\). The ball \(B(x, \epsilon)\) is contained within the rectangle because every component \(x_i\) satisfies \(x_i - \delta < y_i < x_i + \delta\), ensuring \(\| y - x \| < \delta \) and thus \(B(x, \delta) \subset R(x, \delta)\).

Step 4: Concluding Open Set Definition Equivalence

Since a rectangle \(R(x, \delta)\) can contain a ball and a ball can be contained within a rectangle, we have established that for every \(x \in U\) there exists a \(\delta > 0\) ensuring \(R(x, \delta) \subset U\) when \(U\) is open. Conversely, if \(R(x, \delta) \subset U\), a ball \(B(x, \epsilon)\) exists such that \(B(x, \epsilon) \subset U\), confirming that \(U\) is open.

Key Concepts

Open Sets

In the realm of metric spaces, an open set is a versatile concept that helps in understanding topology, especially in Euclidean spaces, like \(\mathbb{R}^n\). Let's break this down:

• An open set contains its boundary, which means if you pick any point \(x\) from the open set \(U\), you can find a small neighborhood or space around \(x\) that also lies completely within \(U\).
• In the context of standard metrics, any point in an open set \(U\) means there's always an open ball around \(x\), with some radius \(\epsilon\) that remains inside \(U\).

The challenge with open sets is comprehending that there's some flexibility and play with how large or small this \(\epsilon\) (the radius of the ball) needs to be. But fundamentally being 'open' means there's no such thing as an isolated edge within the set.

Standard Metric

The standard metric, often experienced in everyday life, is the Euclidean distance. It's how we typically measure distance and how we determine the size of spaces and sets in the Euclidean space \(\mathbb{R}^n\).

• The standard metric is calculated as \(\| x - y \| = \sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + \cdots + (x_n-y_n)^2}\), which is essentially the Pythagorean theorem extended to multiple dimensions.
• This metric helps in forming what's known as 'open balls'. The idea is that within this defined distance, or radius \(\epsilon\), you can create a round area centered at point \(x\) where every point inside maintains a distance lesser than \(\epsilon\) from \(x\).

In essence, the standard metric is a way of quantifying how these open sets behave and interact in space.

Rectangles in Euclidean Space

Rectangles in Euclidean space offer a fascinating alternative to understanding open sets. Unlike open balls, rectangles are defined as Cartesian products of intervals. This makes them slightly box-shaped, rather than circular.

• A rectangle \(R(x, \delta)\) itself is defined by intervals around each coordinate \(x_i\), stretching from \(x_i - \delta\) to \(x_i + \delta\).
• These rectangles are important because they can encompass or be encompassed by open balls by appropriately choosing \(\delta\). For example, by setting \(\delta\) quite small, a rectangle can snugly fit within an open ball.

This property allows rectangles to serve as building blocks for open sets, giving rise to the equivalence between open balls and rectangles in establishing the concept of open sets.

Open Balls

Open balls are an essential concept in comprehending open sets within metric spaces. In the standard metric, an open ball \(B(x, \epsilon)\) creates a perfectly rounded boundary.

• The open ball is defined around a center point \(x\) and extends in every direction with radius \(\epsilon\), such that every point \(y\) within this ball is at a distance less than \(\epsilon\) from \(x\).
• This structure of an open ball ensures that it comfortably nests entirely within an open set \(U\) if \(x \in U\).

These open balls are not only foundational for understanding metric spaces but also facilitate in establishing the neighborhood concept that is vital for defining open sets. They also relate to rectangles in demonstrating how different geometric shapes can be used interchangeably to define openness in \(\mathbb{R}^{n}\).