Summary | Mathematics - Wyatt's Notes

Problem: Is the collection $\tau = \{\emptyset, \{a\}, \{b\}, \{a,b,c\}\}$ a topology on $X = \{a,b,c\}$? Solution: Check axioms: (1) $\emptyset$ and $X$ are in $\tau$. (2) Finite unions: $\{a\} \cup \{b\} = \{a,b\}$, which is NOT in $\tau$. Therefore $\tau$ is not a topology. To fix it, we would need to include $\{a,b\}$.

Problem: Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is continuous with respect to the standard topology. Solution: Let $U$ be an open set in $\mathbb{R}$. $f^{-1}(U) = \{x : x^2 \in U\}$. For any open interval $(a, b)$ with $a \geq 0$, the preimage is $(-\sqrt{b}, -\sqrt{a}) \cup (\sqrt{a}, \sqrt{b})$, which is a union of open intervals (open). For negative intervals, the preimage is empty or $\mathbb{R}$. Since the preimage of any basis element is open, $f$ is continuous.

Problem: Prove that $(0, 1)$ and $(0, \infty)$ are homeomorphic.

Solution: Define $f : (0, 1) \to (0, \infty)$ by $f(x) = x/(1-x)$. This is continuous and bijective with inverse $f^{-1}(y) = y/(1+y)$, also continuous. Hence $(0, 1) \cong (0, \infty)$.

Problem: Is the set $\{1/n : n \in \mathbb{N}\} \cup \{0\}$ compact in $\mathbb{R}$?

Solution: Yes. The set is closed (its only limit point is $0$, which is included) and bounded (contained in $[0, 1]$). By Heine-Borel, it is compact. Alternatively, any open cover contains a neighbourhood of $0$ that covers all but finitely many points.

Problem: Show that any star-shaped subset $S \subseteq \mathbb{R}^n$ is path-connected.

Solution: A set $S$ is star-shaped if there exists $x_0 \in S$ such that for all $x \in S$, the segment $[x_0, x] \subseteq S$. For any $x, y \in S$, define the path $\gamma(t) = (1-2t)x + 2t x_0$ for $t \in [0, 1/2]$ and $\gamma(t) = (2-2t)x_0 + (2t-1)y$ for $t \in [1/2, 1]$. This path is continuous and stays inside $S$, so $S$ is path-connected.

Problem: Compute $\pi_1(S^1)$.

Solution: $\pi_1(S^1) \cong \mathbb{Z}$. The isomorphism maps each loop to its winding number around the circle. This is proved using covering space theory: the exponential map $p : \mathbb{R} \to S^1$, $p(t) = e^{2\pi i t}$, is a covering map. Lifting a loop $\gamma$ in $S^1$ to a path $\tilde\gamma$ in $\mathbb{R}$ gives a unique lift starting at $0$; the endpoint $\tilde\gamma(1)$ is an integer (the winding number), and this defines the isomorphism.

1. Determine whether the set $C = \{f \in C([0,1]) : \|f\|_\infty \leq 1\}$ is compact in the sup-norm topology.
2. Prove that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
3. Show that $\mathbb{R}^2$ and $\mathbb{R}^3$ are not homeomorphic (hint: consider removing a point).
4. Compute $\pi_1(\mathbb{R}P^2)$ using the Seifert-van Kampen theorem.
5. Determine whether the product of two connected spaces is connected.
6. Prove that a metric space is compact if and only if it is complete and totally bounded.
7. Show that the fundamental group of a product space is the direct product of the fundamental groups.
8. Classify all compact connected 2-manifolds up to homeomorphism by their Euler characteristic and orientability.

flowchart TD
    A[11_Summary] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Topology is often called “rubber-sheet geometry” because it studies properties that survive continuous stretching and bending. A coffee mug and a donut are topologically identical because one can be smoothly deformed into the other. The key insight is that topology ignores distances and angles entirely, focusing instead on connectivity: how many holes a space has, whether points can be separated, and whether curves can be shrunk to a point. Compactness captures the idea that “closed and bounded” means no escape to infinity, while connectedness means the space comes in one piece.

Mistake 1: Assuming compactness equals sequential compactness as a rule topological spaces In metric spaces, compactness and sequential compactness are equivalent. as a rule topological spaces, they are not: the space $[0, \omega_1)$ with the order topology is sequentially compact but not compact. Students often assume these concepts are interchangeable without checking whether the space is metrizable.

Mistake 2: Assuming connectedness implies path-connectedness Every path-connected space is connected, but the converse is false. The topologist’s sine curve $\{(x, \sin(1/x)) : x > 0\} \cup \{(0, y) : -1 \leq y \leq 1\}$ is connected but not path-connected. Students frequently use these terms interchangeably, which leads to errors when constructing continuous paths between points.

Mistake 3: Forgetting that the product of Hausdorff spaces is Hausdorff, but the product of normal spaces need not be normal Tychonoff’s theorem guarantees that products of compact spaces are compact, and finite products of Hausdorff spaces are Hausdorff. However, the Sorgenfrey plane (product of two Sorgenfrey lines) is a classic example showing that the product of normal spaces need not be normal. Students often assume normality is preserved under products.

1. Determine whether the following subsets of $\mathbb{R}$ are compact, connected, both, or neither: (a) $[0,1] \cup \{2\}$ (b) $\mathbb{Q} \cap [0,1]$ (c) Cantor set (d) $\{1/n : n \in \mathbb{N}\} \cup \{0\}$.

2. Prove that every compact subset of a Hausdorff space is closed. Why does this fail in non-Hausdorff spaces? Give a counterexample using the cofinite topology.

3. Show that the fundamental group of the figure-eight space $S^1 \vee S^1$ is the free group on two generators $F_2$.

4. Prove that any continuous function $f : S^1 \to \mathbb{R}$ attains its maximum and minimum.

5. Determine whether $[0,1]^\omega$ (the countable product of $[0,1]$ with the product topology) is compact. Justify using Tychonoff’s theorem.

6. Show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ for $n \neq m$ (hint: remove a point and compare fundamental groups or homology groups).

7. Prove that the Sorgenfrey line (lower-limit topology on $\mathbb{R}$) is separable but not second countable. Show that it is completely regular but not normal, and that its square (the Sorgenfrey plane) is not normal.

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