91Ó°ÊÓ

Give counterexamples to the following false statements. (a) The isolated points of a set form a closed set. (b) Every open set contains at least two points. (c) If \(S_{1}\) and \(S_{2}\) are arbitrary sets, then \(\partial\left(S_{1} \cup S_{2}\right)=\partial S_{1} \cup \partial S_{2}\). (d) If \(S_{1}\) and \(S_{2}\) are arbitrary sets, then \(\partial\left(S_{1} \cap S_{2}\right)=\partial S_{1} \cap \partial S_{2}\). (e) The supremum of a bounded nonempty set is the greatest of its limit points. (f) If \(S\) is any set, then \(\partial(\partial S)=\partial S\). (g) If \(S\) is any set, then \(\partial \bar{S}=\partial S\). (h) If \(S_{1}\) and \(S_{2}\) are arbitrary sets, then \(\left(S_{1} \cup S_{2}\right)^{0}=S_{1}^{0} \cup S_{2}^{0}\).

Step by step solution

01

Counterexample 1: Isolated points of a set forming a closed set

Consider the set \(S = \{ 1,2 \} \subset \mathbb{R}\). The isolated points of \(S\) are the set \(S\) itself, which is a closed set. Thus, this statement can be true in some cases.

02

Counterexample 2: Every open set containing at least two points

Consider the open set \(O = \left(0, \frac{1}{2}\right) \subset \mathbb{R}\), where \(O\) has infinitely many points, but not at least two distinct points.

03

Counterexample 3: Boundary of union of two sets

Consider the sets \(S_1 = (0,1)\) and \(S_2 = (1,2)\) in \(\mathbb{R}\). We have \(\partial(S_1 \cup S_2) = \{0, 2\}\), but \(\partial S_1 \cup \partial S_2 = \{0, 1, 2\}\).

04

Counterexample 4: Boundary of the intersection of two sets

Consider the two disjoint open sets \(S_1 = (0,1)\) and \(S_2 = (1,2)\) in \(\mathbb{R}\). We have \(\partial(S_1 \cap S_2) = \emptyset\), but \(\partial S_1 \cap \partial S_2 = \{1\}\).

05

Counterexample 5: The supremum of a bounded nonempty set is the greatest of its limit points

Consider \(S =\{ \frac{1}{n} \mid n \in \mathbb{N} \} \subset \mathbb{R}\). We have \(\sup(S) = 1\) but the only limit point, which is not the greatest, is \(0\).

06

Counterexample 6: Boundary of the boundary of a set equals its boundary

Consider the open set \(S = (0,1) \subset \mathbb{R}\). We have \(\partial S = \{0, 1\}\), but \(\partial (\partial S) = \emptyset\) since the boundary contains isolated points.

07

Counterexample 7: Boundary of the closure of any set equals its boundary

Consider the set \(S = \mathbb{Q} \cap (0,1)\). The closure of this set is \(\bar{S} = [0,1]\). However, \(\partial \bar{S} = \{0,1\}\) and \(\partial S = [0,1]\) are different.

08

Counterexample 8: Interior of the union of two sets equals the union of their interiors

Consider the sets \(S_1 = [0, 1]\) and \(S_2 = \{1\}\). The set \((S_1 \cup S_2)^0 = (0,1)\), but \(S_1^0 \cup S_2^0 = (0,1) \cup \emptyset = (0,1)\), so the statement holds true for these sets.

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Key Concepts

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

Isolated Points of a Set

Isolated points are elements of a set that stand alone, with no immediate neighbors within the set. Consider a simple number line; if a set has a point that is not clustered with others, that is known as an isolated point.

For example, if you have a set consisting of only the numbers 3 and 5, both numbers are isolated points because there's a 'gap' around them with no other members of the set nearby. However, the notion that isolated points must form a closed set is incorrect. A closed set includes all its limit points, and isolated points have no limit points by definition. A set can be closed and consist solely of isolated points like the set \( \{1,2\} \), which is indeed closed, but this doesn't generalize to all sets of isolated points.

Open Set in Real Analysis

An open set is a collection of points where each point has a neighborhood entirely contained within the set. Imagine a stretch of a beach where every point on the beach has surrounding sand that's also part of the beach; there's room to 'move' without hitting the water or the boardwalk. This is the essential idea of an open set.

One common misunderstanding is thinking that an open set must have multiple points. The truth is, an open set can be so small that it doesn't contain more than one point in a specific metric, like the open interval \( (0, \frac{1}{2}) \) which is an infinite set of points between 0 and 0.5 but not including those end points.

Boundary of a Set

The boundary of a set consists of points that are neither in the interior nor the exterior of the set, kind of like the shoreline of that beach from our open set conceptualization. These points can be approached from within and outside the set.

For example, in the sets \( S_1 = (0,1) \) and \( S_2 = (1,2) \) within the real numbers, the boundary of their union isn't simply the union of their boundaries. The boundary comprises the 'edges' of the combined sets, which for \( S_1 \cup S_2 \) are 0 and 2. The individual boundaries include 0 and 1 for \( S_1 \) and 1 and 2 for \( S_2 \), highlighting that boundaries of union sets do not straightforwardly combine.

Closure of a Set

The closure of a set in real analysis is the original set plus its limit points—the smallest closed set containing the original set. It essentially 'seals' any openness by including the boundary points. If we go back to our beach example, it’s like drawing a line in the sand that encompasses all the sand and shoreline; no water, just beach.

Comparing the boundary of a set and the boundary of its closure could yield different results. The boundary of the closure is more about the 'edge' of the enclosed space and can differ from the set's boundary which might not include all the closure's limit points, as seen in the counterexample with the set of rational numbers within the interval (0,1).

Limit Points

Limit points (or accumulation points) are those around which other points in the set crowd, no matter how close you zoom in. For instance, imagine being surrounded by trees in every direction as you take steps in a forest—the trees are like points in the set, and you are a limit point as there are trees arbitrarily close to you whichever way you step.

Contrary to some beliefs, the supremum (or least upper bound) of a set is not always a limit point, and the greatest limit point is not necessarily the supremum. Consider the set \(\{ \frac{1}{n} | n \in \mathbb{N} \}\). Its supremum is 1, which isn't even a member of the set, much less its greatest limit point, which is actually 0.

Interior of a Set

The interior of a set is where you're 'safe' inside the set, away from the boundary. It’s the largest open set contained within the original set. Back to the beach, it would be being surrounded by sand in all directions without touching the water or the boardwalk—the interior is your 'sandbox'.

It's commonly misinterpreted that the interior of the union of two sets is simply the union of their interiors. While sometimes this can be true, and the interiors might merge seamlessly when the sets combine, this isn’t a rule. Depending on the sets and how they interact, the interior of the union might have a different character than the straightforward union of each set’s interior.