Summary

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.

9. 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) $\{1/n : n \in \mathbb{N}\} \cup \{0\}$.

10. 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.

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

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

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

14. 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).

15. 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.