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The Lumenary · Advanced undergraduate

Metric spaces and point-set topology

Name: ______________________

Date: ______________________

You will learn to

Work with metrics, open and closed sets and the axioms of a topology, including sets that are both open and closed, or neither
Prove that ε–δ continuity between metric spaces agrees with the open-set definition
Prove the Heine–Borel theorem, and that continuous images of compact (respectively connected) spaces are compact (connected)
Use compactness and connectedness to tell spaces apart, and decide when a continuous bijection is a homeomorphism
Construct product spaces and quotient spaces such as the circle, torus and Möbius band from intervals and squares

1.
Which of these formulas define a metric on $\mathbb{R}$ ? (Choose all that apply.)
- A. $d(x, y) = |x - y|^2$
- B. $d(x, y) = \min(1, |x - y|)$
- C. $d(x, y) = \dfrac{|x - y|}{1 + |x - y|}$
- D. $d(x, y) = |x^2 - y^2|$
- E. $d(x, y) = |\arctan x - \arctan y|$
2.
With the usual topology on $\mathbb{R}$ , which of these sets are closed? (Choose all that apply.)
- A. $[0, 1]$
- B. $[0, 1)$
- C. $\mathbb{Z}$
- D. $\mathbb{Q}$
- E. $\{1/n \mid n = 1, 2, 3, \dots\}$
- F. $\{0\} \cup \{1/n \mid n = 1, 2, 3, \dots\}$
3.
How many different topologies are there on the two-element set $\{a, b\}$ ?
4.
A student claims: a continuous map $f : \mathbb{R} \to \mathbb{R}$ sends open sets to open sets. Which of these continuous maps are counterexamples? (Choose all that apply.)
- A. $f(x) = x^2$
- B. $f(x) = 0$
- C. $f(x) = 2x + 1$
- D. $f(x) = e^x$
5.
The map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the circle $S^1$ is a continuous bijection but not a homeomorphism. Theorem 14 says a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Which hypothesis of Theorem 14 fails here?
- A.The domain $[0, 2\pi)$ is not compact.
- B.The circle $S^1$ is not Hausdorff.
- C.The map is not really a bijection.
- D.The map is not continuous.
6.
In which of these metric spaces is the given set closed and bounded but not compact? (Choose all that apply.)
- A. $\{x \in \mathbb{Q} \mid 0 \le x \le 2\}$ in $\mathbb{Q}$ with the usual distance
- B. $\mathbb{N}$ itself, with the discrete metric ( $d(m, n) = 1$ for $m \ne n$ )
- C.The closed unit ball of $\ell^2$ (square-summable sequences)
- D. $[0, 1]$ in $\mathbb{R}$
- E.The closed unit disc in $\mathbb{R}^2$
7.
Start with the square $[0, 1] \times [0, 1]$ and glue $(0, y) \sim (1, 1 - y)$ for every $y \in [0, 1]$ , and nothing else. What is the quotient space?
- A.A Möbius band
- B.A cylinder
- C.A torus
- D.A Klein bottle
8.
Which Lean statement says that the continuous image of a compact set is compact? (Here K : Set X and f : X → Y.)
- A.IsCompact K → Continuous f → IsCompact (f '' K)
- B.IsCompact (f '' K) → Continuous f → IsCompact K
- C.∀ K : Set Y, IsCompact K → Continuous f → IsCompact (f ⁻¹' K)
- D.IsClosed K → Continuous f → IsClosed (f '' K)
9.
Write is_topology(points, opens) that returns True exactly when the set opens of frozensets is a topology on the finite set points (a frozenset). Check: every open set is a subset of points; the empty set and points are open; and the union and intersection of any two open sets are open. (Why is checking pairs enough for a finite family?)
Computational — Practice
10.
Removing one point and counting connected pieces proves that which of these pairs are not homeomorphic? (Choose all that apply.)
- A. $[0, 1]$ and the circle $S^1$
- B. $\mathbb{R}$ and $\mathbb{R}^2$
- C. $S^1$ and the sphere $S^2$
- D. $(0, 1)$ and $\mathbb{R}$
- E. $\mathbb{R}^2$ and $\mathbb{R}^3$

Answer key — Metric spaces and point-set topology

1.
The metrics are $\min(1, |x - y|)$ , $|x - y|/(1 + |x - y|)$ and $|\arctan x - \arctan y|$ . The formula $|x - y|^2$ fails the triangle inequality ( $4 > 1 + 1$ ), and $|x^2 - y^2|$ gives distance $0$ between $1$ and $-1$ . The first two metrics have the same open sets as the usual one, which shows that boundedness is not a topological property.
2.
Closed: $[0, 1]$ , $\mathbb{Z}$ and $\{0\} \cup \{1/n\}$ . Not closed: $[0, 1)$ (missing the limit $1$ ), $\mathbb{Q}$ (its closure is $\mathbb{R}$ ) and $\{1/n\}$ (missing the limit $0$ ). In a metric space, $C$ is closed exactly when every convergent sequence in $C$ has its limit in $C$ .
3.
A topology must contain $\varnothing$ and $\{a, b\}$ ; the only question is which of $\{a\}$ , $\{b\}$ to add, and every choice is closed under unions and intersections. So there are $4$ : indiscrete $\{\varnothing, \{a, b\}\}$ , two Sierpiński topologies (adding $\{a\}$ or $\{b\}$ ), and discrete. The lesson's experiment finds $29$ topologies on three points and $355$ on four.
4.
$x \mapsto x^2$ and the constant map are counterexamples: $(-1, 1) \mapsto [0, 1)$ and $\mathbb{R} \mapsto \{0\}$ . Continuity pulls open sets back; it does not push them forward. Maps that do push open sets forward are called open maps — $2x + 1$ and $e^x$ are examples.
5.
The domain is not compact: $[0, 2\pi)$ is bounded but not closed, so Heine–Borel says it is not compact. Everything else holds. In fact no homeomorphism $[0, 2\pi) \cong S^1$ exists at all, since $S^1$ is compact and compactness is a topological property; the lesson's Lean file proves this (ico_not_homeomorphic_circle).
6.
The first three are closed and bounded but not compact; $[0, 1]$ and the closed disc are compact. In a general metric space, compactness is equivalent to complete and totally bounded: $\mathbb{Q} \cap [0, 2]$ is not complete, and the other two are not totally bounded (infinitely many points at a fixed positive distance from each other).
7.
Gluing one pair of opposite edges gives a cylinder when the edges are matched directly, $(0, y) \sim (1, y)$ , and a Möbius band when they are matched with a flip, $(0, y) \sim (1, 1 - y)$ . Its boundary is a single circle, made of the top and bottom edges joined end to end.
8.
The statement is IsCompact K → Continuous f → IsCompact (f '' K). The other three are false, as shown by a constant map or by $e^x$ . The lesson proves the correct statement from the open-cover definition: pull the cover back along f, take a finite subcover of K, and push it forward.
9.
```
def is_topology(points, opens):
    if frozenset() not in opens or points not in opens:
        return False
    if any(not u <= points for u in opens):
        return False
    for u in opens:
        for v in opens:
            if u | v not in opens or u & v not in opens:
                return False
    return True
```
For a finite family, every union is a union of finitely many sets, and closure under unions of two sets gives closure under unions of any finite number by induction: $U_1 \cup \dots \cup U_k = (U_1 \cup \dots \cup U_{k-1}) \cup U_k$ . The same holds for intersections. For infinite families, closure under arbitrary unions is a genuinely stronger condition.
10.
One-point removal distinguishes $[0, 1]$ from $S^1$ and $\mathbb{R}$ from $\mathbb{R}^2$ . It cannot distinguish $S^1$ from $S^2$ (removing two points can), and it cannot distinguish $\mathbb{R}^2$ from $\mathbb{R}^3$ — that needs the fundamental group of the next lesson. $(0, 1)$ and $\mathbb{R}$ are homeomorphic, so nothing distinguishes them.