3. Topology of ℝ

“The essence of topology is not shape, but closeness.”

Topology studies the structure of spaces based on nearness rather than rigid measurements. While full topology encompasses exotic objects like Möbius strips and coffee-cup morphisms, this chapter will focus on the topology of the real number line, the foundation of calculus and analysis.

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3.1 Intervals and Neighborhoods

In $\mathbb{R}$, the basic topological structure is defined by open intervals:

$$ (a, b) = { x \in \mathbb{R} \mid a < x < b } $$

These intervals represent points “near” a center without touching the endpoints. A neighborhood of a point $x \in \mathbb{R}$ is any open interval containing $x$:

$$ \text{A neighborhood of } x: \quad (x - \varepsilon, x + \varepsilon) $$

This idea is central to limits, continuity, and convergence.

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3.2 Open and Closed Sets

• A set $A \subseteq \mathbb{R}$ is open if for every point $x \in A$, there exists $\varepsilon > 0$ such that $(x - \varepsilon, x + \varepsilon) \subseteq A$.

• A set $B \subseteq \mathbb{R}$ is closed if it contains all of its limit points (see below).

Some sets are both ($\mathbb{R}$); some are neither.

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3.3 Interior, Exterior, and Boundary Points

• Interior point: A point $x \in A$ is interior if there exists a neighborhood of $x$ entirely within $A$.

• Exterior point: A point is exterior to $A$ if it’s interior to the complement $\mathbb{R} \setminus A$.

• Boundary point: A point $x$ is a boundary point of $A$ if every neighborhood around $x$ intersects both $A$ and $\mathbb{R} \setminus A$.

Example:

$$ A = (0, 1) \Rightarrow \text{Interior: } (0,1),; \text{Boundary: } {0,1},; \text{Exterior: } \mathbb{R} \setminus [0,1] $$

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3.4 Adherent and Accumulation Points

• Adherent point of $A$: A point $x \in \mathbb{R}$ such that every neighborhood of $x$ intersects $A$. It may or may not belong to $A$.

• Accumulation point: A stricter case: every neighborhood of $x$ contains another point of $A$. (i.e., $x$ is a limit of a sequence in $A \setminus {x}$).

Example:

$$ A = { 1/n \mid n \in \mathbb{N} } \Rightarrow 0 \text{ is an accumulation point} $$

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3.5 Compactness (In ℝ)

A set $A \subseteq \mathbb{R}$ is compact if it is:

• Closed
• Bounded

This is the Heine–Borel Theorem in ℝ:

A subset of $\mathbb{R}$ is compact $\iff$ it is closed and bounded.

Compact sets have beautiful consequences: every sequence in them has a convergent subsequence. This is essential for understanding limits and continuity.

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This foundational topology allows us to define limits, continuity, and later, the derivative with rigorous precision. Welcome to the soft geometry of the real line.