Problem 2 Of what category is the set of a... [FREE SOLUTION]

91影视

Introductory Functional Analysis with Applications

Erwin Kreyszig

$Math Studyset 91影视 Explanations$ Math

1978 Edition

Chapter 4: Problem 2

Of what category is the set of all integers $(a)$ in $\mathbf{R},(b)$ in itself (taken with the metric induced from $\mathbf{R}$ )?

Short Answer

Expert verified

(a) Closed in $ \mathbf{R} $, (b) Clopen in $ \mathbb{Z} $.

Step by step solution

Understanding the Context

We are given a set of all integers $ \mathbb{Z} $ and we need to determine its category both in the larger space of real numbers $ \mathbf{R} $ and within itself using the Euclidean metric. The Euclidean metric is the standard distance measure in real numbers, defined as $ d(x, y) = |x - y| $.

Determining Category in $ \mathbf{R} $

A set is closed in $ \mathbf{R} $ if it contains all its limit points. In $ \mathbf{R} $, the set $ \mathbb{Z} $ does not contain any limit points because the distance between any two integers is at least 1, so no sequence of integers converges to a non-integer limit. Therefore, $ \mathbb{Z} $ is closed in $ \mathbf{R} $.

Determining Category in Itself

When considering $ \mathbb{Z} $ within itself using the metric induced from $ \mathbf{R} $, the topology is discrete. This means every single point is isolated, and every subset is open by definition. Therefore, since $ \mathbb{Z} $ is the entire space, it is trivially both open and closed (clopen) in itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Set

The integer set, denoted as $ \mathbb{Z} $, is a fundamental concept in mathematics. It includes all positive and negative whole numbers as well as zero. Some of its essential properties include:

The integer set is countably infinite, meaning its elements can be listed in a sequence: ..., -3, -2, -1, 0, 1, 2, 3, ...
It is a subset of the real numbers $ \mathbf{R} $, but unlike the real numbers, it does not include fractions or decimals.
In a metric space, the integers are often examined under different topologies to understand their properties in various contexts.

Understanding the behavior of $ \mathbb{Z} $ under the Euclidean metric helps in comprehending its closed and discrete nature in different metric spaces.

Closed Set

In mathematics, a closed set in a metric space is one that includes all its limit points. When we look at the integer set $ \mathbb{Z} $ in the context of $ \mathbf{R} $, we need to examine its closure properties.

Limit points are points to which a sequence of points in the set converges. In $ \mathbb{Z} $, sequences that could converge to a non-integer limit do not exist within the integers.
The gap between any two consecutive integers is always at least 1, ensuring no integer sequence in $ \mathbb{Z} $ converges outside itself.
Thus, these properties confirm that $ \mathbb{Z} $ is a closed set in $ \mathbf{R} $.

This definition means that in the larger real number space, $ \mathbb{Z} $ is explicitly closed because it does not approach any non-integer points.

Euclidean Metric

The Euclidean metric is a standard way to measure the distance between two points in $ \mathbf{R} $. It is defined as $ d(x, y) = |x - y| $. Here's why it's important when discussing integer sets:

This metric determines how far apart each integer is from another in $ \mathbb{Z} $.
In the integer set, the distance between two consecutive numbers like 1 and 2 is exactly 1.
No two different integers are closer to each other than this fixed minimum distance.

The Euclidean metric helps maintain the concept that integers are isolated points in $ \mathbf{R} $, supporting their closed nature in the real number metric space.

Discrete Topology

Discrete topology is a concept where every subset of a space is considered both open and closed ("clopen"). When we examine $ \mathbb{Z} $ as a standalone metric space with the metric inherited from $ \mathbf{R} $:

Each integer is isolated from all others, meaning it is "open" whether we look at it by itself or as part of $ \mathbb{Z} $.
As $ \mathbb{Z} $ contains all its elements considered as points, it is both open and closed in itself.
This inherent property gives $ \mathbb{Z} $ a discrete topology, where its structure makes it flexible in how subsets are considered.

In practical terms, understanding discrete topology allows us to treat $ \mathbb{Z} $ with simplicity, as there is no further breaking down of elements beyond their definition as discrete whole numbers.

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91影视

Of what category is the set of all integers \((a)\) in \(\mathbf{R},(b)\) in itself (taken with the metric induced from \(\mathbf{R}\) )?

Short Answer

Step by step solution

Understanding the Context

Determining Category in \( \mathbf{R} \)

Determining Category in Itself

Key Concepts

Integer Set

Closed Set

Euclidean Metric

Discrete Topology

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