1. Compactness of the Sets:

Set 1: $[0, 1] \cup [2, 3]$ ⊂ $\mathbb{R}$

• Bounded: The set is bounded because all elements lie between 0 and 3.
• Closed: The set is the union of two closed intervals $[0, 1]$ and $[2, 3]$, which are both closed in $\mathbb{R}$.

Since the set is both closed and bounded, it is compact in $\mathbb{R}$.

Set 2: $\{x \in \mathbb{R} : 0 \leq x \leq 1 \text{ and } x \text{ is irrational}\}$

• Bounded: The set is bounded because all elements lie between 0 and 1.
• Closed: The set is not closed. This is because it does not contain all its limit points. For example, the sequence of irrational numbers approaching a rational number (like $x = 0.5$) does not converge to an element within the set.

Since the set is bounded but not closed, it is not compact in $\mathbb{R}$.

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2. Closed and Boundedness of the Set:

Consider the set $[−1,0) \cup (0,1]$:

• Bounded: The set is bounded because all elements lie between $-1$ and $1$.
• Closed: The set is not closed. This is because it does not include the endpoints $0$ and $-1$, which are limit points of the set.

Since the set is bounded but not closed, it is not compact.

Do you have any questions or need more details?

5 Related Questions:

1. How do you define compactness in a general topological space?
2. What is an example of a set that is closed but not bounded in $\mathbb{R}$?
3. Can a set be compact if it is only closed but not bounded?
4. What are the properties of open sets in terms of compactness?
5. How do irrational numbers impact the closure of a set?

Tip:

When checking for compactness, always verify if the set is both closed and bounded, as these are the key criteria in $\mathbb{R}$.