Question

Suppose that a sequence \(U_{1}, U_{2}, \ldots\) of open sets covers a compact set \(S\). Without using the Heine-Borel theorem, show that \(S \subset \bigcup_{m=1}^{N} U_{m}\) for some \(N\).

Short answer

Question: Given a compact set \(S\) and a sequence of open sets \((U_1, U_2, ...)\) that covers \(S\), show that there exists a positive integer \(N\) such that \(S \subset \bigcup_{m=1}^{N} U_{m}\), without using the Heine-Borel theorem. Answer: To show the existence of such an integer \(N\), we use the definition of compactness stating that a set is compact if every open cover has a finite subcover. We know that the finite subcover \((U_{i_1}, U_{i_2}, \ldots, U_{i_N})\) covers \(S\), and letting \(M = \max\{i_1, i_2,\dots, i_N\}\), we get that \(S \subset \bigcup_{m=1}^M U_m\). Using \(N = M\), we have found the required set satisfying the given condition.

Step-by-step solution

Define the compact set and open cover

We are given a compact set \(S\) and a sequence \((U_1, U_2, \ldots)\) of open sets that covers \(S\). This means that for every element \(s \in S\), there exists an open set \(U_i\) in the sequence such that \(s \in U_i\).

Use the definition of compactness

The definition of compactness states that a set \(S\) is compact if every open cover has a finite subcover. This means that there exists a finite collection of sets \((U_{i_1}, U_{i_2}, \ldots, U_{i_N})\) from our sequence that still covers the whole set \(S\). We will represent this finite subcover as follows: \(S \subset \bigcup_{n=1}^N U_{i_n}\).

Find the required set

We need to show that there exists some \(N\) such that \(S \subset \bigcup_{m=1}^{N} U_{m}\). From step 2, we already have a finite collection of sets \((U_{i_1}, U_{i_2}, \ldots, U_{i_N})\) that covers \(S\), so we can write \(S \subset \bigcup_{n=1}^N U_{i_n}\). Let \(M = \max\{i_1, i_2,\dots, i_N\}\). Then, since for each \(n\) we have \(i_n\le M\), we can deduce that \(S \subset \bigcup_{n=1}^N U_{i_n} \subseteq \bigcup_{m=1}^M U_m\). Thus, we have found the required set which confirms that \(S \subset \bigcup_{m=1}^M U_m\) for some \(M\). We can use \(N = M\) to fulfill the condition given in the exercise.

Key concepts

Open Sets

In topology, an open set is a fundamental concept that helps us understand how sets can arrange or separate in space. Simply put, an open set in a given space does not include its boundary. Imagine you have a circle drawn on a piece of paper: an open circle would include all points inside but not the edge itself. In a broader sense, here's how to think about open sets:

• **They allow wiggle room:** Any point inside an open set is surrounded by other points from the set, without any gap.
• **Local availability:** For any point within the set, it's possible to find a nearby neighborhood completely contained in the set.
• **Common in different spaces:** This concept is not limited to familiar spaces like the number line or a plane, but extends to more complex structures.

These simple properties make open sets quite useful in mathematical studies, especially in defining and understanding the idea of continuity and limits.

Open Cover

Open covers are collections of open sets that come together to cover a given set. For example, say we have a set that we want to cover with open sets, the open cover will encompass the entire set using potentially overlapping open sets. This involves coordinating multiple sets so that their union (or combination) completely contains the original set. Here's why they are important:

• **Complete coverage:** An open cover ensures every element in the set is included in at least one of the open sets.
• **Variety in size:** Open covers can include sets of varying sizes and shapes, offering flexibility in coverage.
• **Infinite possibilities:** The cover doesn't have to be finite; it can consist of an infinite number of open sets.

In essence, open covers connect the idea of open sets to larger sections of space, often essential in compactness and continuity discussions in topology.

Finite Subcover

A finite subcover is a critical concept when dealing with compact sets. When you have an open cover of a compact set, such as a series of open sets, a finite subcover is a selection of those open sets that still covers the entire original set but uses only a finite number of them. This is central to understanding compactness:

• **Simplifying infinite to finite:** Every open cover of a compact set can be reduced to a finite subcover, showcasing the efficiency of compact sets.
• **Ensures manageability:** By finding a finite subcover, working with and analyzing the set becomes much more straightforward.
• **Proof of compactness:** The ability to derive a finite subcover from an open cover is a defining property of compact sets.

This trait makes compact sets particularly important in mathematical analysis, as they behave nicely under many operations and allow generalization of finiteness in infinite scenarios.