In real analysis, the concepts of open, closed, and compact sets are fundamental in understanding the topology of the real number line $\mathbb{R}$ and higher-dimensional spaces. Here's a detailed explanation:

1. Open Sets

An open set is a set in which, for every point within the set, there is a small neighborhood around that point that is also entirely contained within the set. Intuitively, an open set does not include its boundary points.

Formal Definition: A set $U \subseteq \mathbb{R}$ is called open if, for every $x \in U$, there exists an $\epsilon > 0$ such that the interval $(x - \epsilon, x + \epsilon) \subseteq U$.

Example:

• The set $U = (1, 2)$ is open because for any point $x \in (1, 2)$, you can find a small interval around $x$ that is still within $(1, 2)$.
• The entire real line $\mathbb{R}$ is an open set.
• The empty set $\emptyset$ is also considered open.

2. Closed Sets

A closed set is a set that contains all its boundary points. Equivalently, a set is closed if it contains all its limit points or if its complement is open.

Formal Definition: A set $C \subseteq \mathbb{R}$ is called closed if it contains all its limit points, or equivalently, if its complement $\mathbb{R} \setminus C$ is open.

Example:

• The set $C = [1, 2]$ is closed because it includes the boundary points $1$ and $2$.
• The set $\mathbb{R}$ is closed because it contains all its boundary points, and its complement is the empty set, which is open.
• The empty set $\emptyset$ is also closed.

3. Compact Sets

A compact set is a set that is both closed and bounded. In $\mathbb{R}$, a set is compact if every open cover has a finite subcover (this is known as the Heine-Borel Theorem in $\mathbb{R}$).

Formal Definition: A set $K \subseteq \mathbb{R}$ is compact if it is closed and bounded. Alternatively, $K$ is compact if for every open cover $\{U_i\}_{i \in I}$ of $K$, there exists a finite subcover $\{U_{i_1}, U_{i_2}, \dots, U_{i_n}\}$ that still covers $K$.

Example:

• The set $K = [1, 2]$ is compact because it is closed (contains its boundary points) and bounded (contained within a finite interval).
• The set $K = [0, 1]$ is another example of a compact set in $\mathbb{R}$.
• The set $K = \mathbb{R}$ is not compact because it is not bounded.

Visual Intuition

• Open Set: Think of an interval like $(1, 2)$. You can approach but never touch the endpoints 1 and 2.
• Closed Set: Imagine the interval $[1, 2]$, where you can touch the endpoints.
• Compact Set: It's like a closed interval that has a maximum and minimum point, such as $[1, 2]$. It's "contained" in a finite region.

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Would you like more details or have any questions on this topic?

Here are 5 questions that extend this discussion:

1. How do you prove that the union of open sets is open?
2. What are examples of sets that are neither open nor closed?
3. How does the concept of compactness change in higher dimensions?
4. Can a set be open and closed simultaneously? What are such sets called?
5. How is compactness related to sequences in real analysis?

Tip: To understand compact sets deeply, study the Heine-Borel Theorem, which characterizes compact sets in $\mathbb{R}^n$ as those that are closed and bounded.