Metric spaces and point-set topology

Which of these formulas define a metric on $\mathbb{R}$? (Choose all that apply.)

With the usual topology on $\mathbb{R}$, which of these sets are closed? (Choose all that apply.)

How many different topologies are there on the two-element set $\{a, b\}$?

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.)

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?

In which of these metric spaces is the given set closed and bounded but not compact? (Choose all that apply.)

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?

Which Lean statement says that the continuous image of a compact set is compact? (Here K : Set X and f : X → Y.)

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?)

Removing one point and counting connected pieces proves that which of these pairs are not homeomorphic? (Choose all that apply.)