• second countability
• metrizability
• embedding theorems
• bump functions

The Long Line, Second Countability

1. What is Second Countability?

A topological space is second-countable if its topology has a countable base.

To put it simply: you can describe every open set in the space using a “construction kit” that contains only a countable number of basic shapes. For example, the standard real line is second-countable because every open interval can be built using intervals with rational endpoints, and the set of pairs of rational numbers is countable.

Why it matters:

• Metric Spaces: For a metric space, being second-countable is equivalent to being separable (having a countable dense subset).
• Embeddings: Second-countable manifolds can be embedded into higher-dimensional Euclidean spaces.
• Compactness: It helps ensure that concepts like “compactness” and “sequential compactness” behave predictably.

2. The Long Line ($\mathbf{L}$)

The Long Line is a topological space that is “longer” than the real line $\mathbb{R}$. While the real line is built by placing unit intervals side-by-side indexed by the integers, the Long Line is built by placing unit intervals side-by-side indexed by the first uncountable ordinal, .

Key Properties:

• Locally Euclidean: If you zoom in on any point, it looks exactly like a piece of the standard real line. It is a 1-dimensional manifold.
• Not Second-Countable: Because there are uncountably many “units” glued together, you cannot describe its topology with a countable base.
• Normal but not Metric: It is a normal space, but because it isn’t second-countable, it cannot be metrized. You can’t define a standard distance that works for the whole line.
• Path Connected: You can draw a continuous path between any two points.

3. Comparison: The Real Line vs. The Long Line

Why should you care?

The Long Line is the “Boogeyman” of topology. It warns mathematicians that local properties (looking like at every point) do not always dictate global properties (being able to measure the whole thing with a ruler). It proves that you can have a space that is perfectly smooth and continuous, yet so vast that it breaks the fundamental rules of distance.

1. The Urysohn Metrization Theorem

Urysohn’s theorem provides a sufficient condition for a topological space to be metrizable. It is a “top-down” approach: if a space is small enough (second-countable) and separated enough (regular), it must be metrizable.

Theorem: Every second-countable, regular $T_2$ space is metrizable.

• Regular ($T_3$): For every closed set $F \subset X$ and point $x \notin F$, there exist disjoint open sets $U$ and $V$ such that $x \in U$ and $F \subset V$.
• Second-countable: The topology $\mathcal{T}$ has a countable base $\mathcal{B} = \{B_n\}_{n \in \mathbb{N}}$.

2. The Bing-Nagata-Smirnov Theorem

This theorem provides the necessary and sufficient (if and only if) conditions for metrizability, covering spaces that Urysohn’s theorem might miss.

Theorem: A topological space $X$ is metrizable if and only if it is regular, Hausdorff, and has a $\sigma$-locally finite base.

• $\sigma$-locally finite base: A base $\mathcal{B}$ is $\sigma$-locally finite if $\mathcal{B} = \bigcup_{n \in \mathbb{N}} \mathcal{B}_n$, where each collection $\mathcal{B}_n$ is locally finite.
• Local Finiteness: Each point $x \in X$ has a neighborhood $W$ such that the set $\{B \in \mathcal{B}_n : B \cap W \neq \emptyset\}$ is finite.

3. Why Metrizability Matters

In a metrizable space $(X, d)$, the topology behaves with a specific “rigidity” that general spaces lack:

• Sequences: A point $x$ is in the closure of $A$ if and only if there is a sequence $(x_n) \subseteq A$ such that $x_n \to x$. In non-metrizable spaces, this is not always true, requiring the use of nets $(x_\alpha)_{\alpha \in A}$.
• Distance Function: We can define a function $d: X \times X \to [0, \infty)$ that satisfies the triangle inequality: $$d(x, z) \leq d(x, y) + d(y, z)$$
• Compactness: In any metric space, Compactness, Sequential Compactness, and Countable Compactness are all equivalent.

4. Why the Long Line $L$ Fails

The Long Line is the quintessential example of a non-metrizable manifold.

• Local Metrizability: Every point $p \in L$ has a neighborhood $U$ such that $U \cong (0, 1)$. Since $(0, 1)$ is metrizable, $L$ is locally metrizable.
• Global Failure: The Long Line is constructed as $L = \omega_1 \times [0, 1)$ (where $\omega_1$ is the first uncountable ordinal).
• The Contradiction: $L$ is sequentially compact but not compact. In a metrizable space, these two properties must coincide. Therefore, $L$ cannot be metrized.
• Size: $L$ is "

1. The Core Idea: Manifolds as Metric Spaces

By definition, a manifold $M$ is locally Euclidean. This means every point $p \in M$ has a neighborhood $U$ such that there exists a homeomorphism $\phi: U \to V$, where $V$ is an open subset of $\mathbb{R}^n$.

If a manifold is compact, several “niceness” properties converge: * It is automatically second-countable. * It is Hausdorff ($T_2$) and normal ($T_4$). * Because it is second-countable and regular, the Urysohn Metrization Theorem tells us that every compact manifold is metrizable.

The takeaway: Every compact manifold is a metric space, meaning there exists a metric $d: M \times M \to [0, \infty)$ that induces the manifold’s topology.

2. Embedding into Euclidean Space $\mathbb{R}^N$

The most common metric space target is $\mathbb{R}^N$. The Whitney Embedding Theorem provides the upper bound for the dimension $N$ required to fit an $n$-dimensional manifold.

• Weak Whitney Theorem: Any smooth, compact $n$-dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$.
• Strong Whitney Theorem: Any smooth, compact $n$-dimensional manifold can be embedded in $\mathbb{R}^{2n}$.

For example: * The circle $S^1$ (where $n=1$) embeds into $\mathbb{R}^2$. * The Klein Bottle (where $n=2$) cannot embed into $\mathbb{R}^3$ without self-intersection, but it embeds into $\mathbb{R}^4$.

3. The Mechanism: Partition of Unity

To construct a global embedding, we use a partition of unity. This allows us to “glue” local coordinate maps together.

1. Cover $M$ with a finite collection of coordinate patches $\{U_i\}_{i=1}^k$ and homeomorphisms $\phi_i: U_i \to \mathbb{R}^n$.
2. Define a set of smooth “bump functions” $\psi_i: M \to [0, 1]$ such that the support of $\psi_i$ is contained in $U_i$ and $\sum \psi_i(x) = 1$ for all $x \in M$.
3. Construct the global map $f: M \to \mathbb{R}^{k(n+1)}$ by: $$f(x) = (\psi_1(x)\phi_1(x), \dots, \psi_k(x)\phi_k(x), \psi_1(x), \dots, \psi_k(x))$$

Since $M$ is compact and $f$ is a continuous injection into a Hausdorff space, $f$ is a homeomorphism onto its image.

4. Why the Long Line $L$ Fails

The Long Line is a $1$-dimensional manifold, but it cannot be embedded into any metric space $\mathbb{R}^N$.

• Dimension vs. Size: While $L$ is locally $1$-dimensional, it is not second-countable.
• The Euclidean Barrier: Any subspace of $\mathbb{R}^N$ must be second-countable (because $\mathbb{R}^N$ has a countable base of rational balls).
• Since $L$ is “too long” (its length is indexed by the first uncountable ordinal $\omega_1$), it cannot be contained within the countable structure of Euclidean space.

Summary Table

References

1. References in Michael Spivak

Book: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus

In this text, Spivak introduces bump functions specifically to build a Partition of Unity, which he then uses to define integration on manifolds.

• Theorem 3-11 (The Existence of Bump Functions): Found in Chapter 3 (Integration). Spivak constructs a smooth function $f: \mathbb{R}^n \to \mathbb{R}$ such that $f(x) > 0$ for $|x| < a$ and $f(x) = 0$ for $|x| \geq a$.
• Construction: He starts with the classic non-analytic smooth function: $$\lambda(x) = \begin{cases} e^{-1/x^2} & x > 0 \\ 0 & x \leq 0 \end{cases}$$
• Partition of Unity: This is covered in the subsequent pages of Chapter 3. He proves that for any open cover $\mathcal{O}$ of a closed rectangle $A$, there exist smooth functions $\phi_i$ subordinate to the cover such that $\sum \phi_i = 1$.

2. References in Guillemin & Pollack

Book: Differential Topology (Victor Guillemin and Alan Pollack)

Guillemin and Pollack (often referred to as G&P) use bump functions more explicitly for embeddings and the Whitney Embedding Theorem.

• Chapter 1, Section 6 (Partition of Unity): G&P introduce the “Bump Function” as a technical Lemma.
• The Lemma: They prove that for any closed set $A$ and open set $U$ containing $A$ ($A \subset U$), there exists a smooth function $\rho$ such that $0 \leq \rho \leq 1$, where $\rho = 1$ on $A$ and $\text{support}(\rho) \subset U$.
• Applications: In the same section, they use these functions to:
  1. Extend local maps $f: U \to \mathbb{R}^n$ to global maps.
  2. Construct the embedding of a compact manifold $X$ into $\mathbb{R}^N$.

3. Comparison of Approach

Why this matters for your Embedding

As mentioned in the previous sections, to embed a manifold into a metric space, you need a way to “turn off” a coordinate map $\phi_i$ before you leave its domain $U_i$.

If you have a map $\phi_i: U_i \to \mathbb{R}^n$, the product $\psi_i(x)\phi_i(x)$ (where $\psi_i$ is a bump function) is well-defined on the entire manifold $M$. Without the bump function, the map would “blow up” or be undefined outside of $U_i$.