Exercise 7.0. This is an unseen exercise in Applied Topology. In the diagram below, each letter of the English alphabet is drawn as a union of straight line segments and arcs.

Some letters are homeomorphic: for example, C $\cong$ J, both are homeomorphic to a closed interval. Consider such homeomorphisms to be geometrically obvious.

Some letters are not homeomorphic: here is a topological property that can distinguish them. If $X$ is a topological space, call $p\in X$ a point of connectivity $k$ if $X\setminus \{p\}$ has exactly $k$ connected components. The following is easy to prove: any homeomorphism $X\stackrel{\sim }{\to }Y$ maps a point of connectivity $k$ to a point of connectivity $k.$ Hence, for each $k,$ the number of points of connectivity $k$ is a topological property. Example:

O $\ncong$ P: O has no points of connectivity 2 but P has them;

T $\ncong$ O and T $\ncong$ P: T has a point of connectivity 3 while O, P have no such points.

CHALLENGE. Sort the letters into homeomorphism classes. You should have 9 classes.

Class 1: have a point of connectivity 4: K X

Class 2: has two points of connectivity 3: H

Class 3: one point of connectivity 3, three points of connectivity 1: E F T Y

Class 4: one point of connectivity 3, infinitely many points of connectivity 1: Q R

Class 5: one point of connectivity 2: B

Class 6: the set of points of connectivity 2 is disconnected: A

Class 7: the set of points of connectivity 2 is connected: P

Class 8: intervals — two points of connectivity 1, all other points are of connectivity 2: C G I J L M N S U V W Z

Class 9: circles — all points are of connectivity 1: D O

Exercise 7.1. Consider the topological space $\mathbb{Q}$ which is the set of all rational numbers, viewed as a subspace of the Euclidean real line $\mathbb{R}.$

• 1. Is $\mathbb{Q}$ Hausdorff? Is $\mathbb{Q}$ compact? Justify your answer.

• 2. Show that the topology on $\mathbb{Q}$ is not discrete.

• 3. A topological space $X$ is called totally disconnected if every non-empty connected subset of $X$ is a singleton. Show that $\mathbb{Q}$ is totally disconnected.