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 \(X\xrightarrow {\sim } Y\) 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:

Class 2:

Class 3:

Class 4:

Class 5:

Class 6:

Class 7:

Class 8:

Class 9:

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 \(\RR .\)

• 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.