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Chapter 13: Problem 3

Show that every finite subset of a metric space is compact.

Short Answer

Expert verified
Every finite subset of a metric space is compact, as every open cover has a finite subcover.

Step by step solution

01

Understanding Compactness in Metric Space

In a metric space, a set is compact if every open cover of the set has a finite subcover. This means that if you have a collection of open sets that altogether cover the set, you can select a finite number of those open sets that still cover the set. Our task is to show that this is true for any finite subset within a metric space.
02

Defining the Finite Subset

Let's consider a finite subset of a metric space, denoted as \( A = \{ a_1, a_2, ..., a_n \} \). In this step, we are defining our set \( A \), which contains a finite number of elements \( n \).
03

Considering an Open Cover

An open cover of the set \( A \) is a collection of open sets such that each element of \( A \) is contained in at least one of these open sets. Let's denote the open cover by \( \{ U_i \}_{i \in I} \), where \( I \) is an index set.
04

Selecting a Finite Subcover

Since \( A \) is a finite set, we can choose open sets from the open cover \( \{ U_i \}_{i \in I} \) for each element \( a_j \) in \( A \), making sure \( a_j \in U_{i_j} \). Thus, \( \{ U_{i_1}, U_{i_2}, ..., U_{i_n} \} \) is a finite subset of our original open cover that still covers \( A \).
05

Verifying the Compactness Criterion

The constructed set \( \{ U_{i_1}, U_{i_2}, ..., U_{i_n} \} \) is a finite subcover, as it contains open sets that collectively cover the entire set \( A \). This confirms that every open cover of \( A \) has a finite subcover, fulfilling the compactness criterion.

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Key Concepts

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

finite subsets
A finite subset consists of a limited number of elements, none of which can be divided further in terms of mathematical consideration. When we talk about finite subsets in metric spaces, we are referring to a particular set within a space that is measurable by distances between points. Examples of finite subsets might include
  • A small group of points spread across a metric space.
  • A finite number of moments or events placed over a time sequence.
Finite subsets have a distinct advantage when proving compactness because each element can be paired or connected quite feasibly with only a short list of open sets. This quality simplifies identifying the finite subcover that proves compactness.
open covers
An open cover in the context of metric spaces refers to a collection of open sets that, together, encompass a subset of the space completely. These open sets are like blankets that cover each point in the subset.

For a subset to be covered by open sets means every single element of that subset is contained in at least one of these open covers. This coverage might involve
  • Basic intervals on a real line or surface.
  • Balls or circles surrounding points in higher dimensional spaces.
The purpose of identifying an open cover is to demonstrate whether it is possible to select a smaller number of these open sets that still serve the function of covering the subset, which is crucial for compactness.
metric spaces
Metric spaces are a type of mathematical landscape where distances can be precisely measured. They are defined using a metric or a distance function, which helps in specifying how far apart any two points in that space are. Metric spaces encompass a wide range of configurations, including
  • The set of real numbers, where the usual way to measure distance is with absolute difference.
  • 2-dimensional surfaces like planes, where distances are calculated using the Euclidean formula.
The structure of metric spaces provides the logical framework needed to discuss concepts like continuity, limits, and compactness, applying these universally across varying types of spaces.
finite subcovers
When an open cover of a subset has a finite subcover, it signifies that you can select a limited number of open sets from the original cover and they will still completely encapsulate the subset. Finite subcovers are vital for demonstrating that a subset is compact.

Consider a set that is covered by numerous open intervals. If you can pinpoint and select only a handful of those intervals that still manage to cover every point of your set, then you have effectively found a finite subcover. By demonstrating the existence of a finite subcover, we are able to confirm that the entire set abides by the rule of compactness, where infinite possibilities are reduced to a countable, manageable subset of open sets.

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Most popular questions from this chapter

(a) Show that a set \(A\) in a metric space \(X\) is nowhere dense if and only if \(\bar{A}\) has empty interior. (b) Show that a closed set in a metric space \(X\) is nowhere dense if and only if its complement is dense in \(X\).

Use the Heine-Borel property to prove that if \(f: X \rightarrow Y\) is continuous and \(X\) is compact, then the set \(f(X)\) is compact in \(Y\).

Let \((X, d)\) and \((Y, e)\) be metric spaces and let \(f\) be a continuous function mapping \(X\) onto \(Y\). (a) If \(X\) is separable, must \(Y\) be separable? (b) If \(X\) is complete, must \(Y\) be complete? (c) Is separability a topological property? Is completeness? (d) Do the answers to (a) and (b) change if \(f\) is an isometry?

Show that the function \(f: \ell_{2} \rightarrow \ell_{2}\) defined on Hilbert space by specifying $$ f\left(x_{1}, x_{2}, x_{3}, x_{4}, \ldots\right)=\left(0, x_{1}, x_{2}, x_{3}, \ldots\right) $$ for every \(\left(x_{1}, x_{2}, x_{3}, x_{4}, \ldots\right) \in \ell_{2}\) is an isometric map of \(\ell_{2}\) to a nowhere dense subset.

The mathematician Herman Weyl is credited with joking that a "compact city is a city that can be guarded by a finite number of arbitrarily nearsighted policemen." Explain.
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