\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \(\require {textcomp}\) \(\newcommand {\RR }{\mathbb {R}} \) \(\DeclareMathOperator {\Cl }{\mathrm {Cl}} \) \(\DeclareMathOperator {\Int }{\mathrm {Int}} \) \(\renewcommand {\qedhere }{} \)

Week 7 Exercises

Version 2024/11/20. These exercises in PDF To other course material

  • 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.

    (-tikz- diagram)

    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:

    (-tikz- diagram)

    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.

[Answers to these exercises]

Version 2024/11/20. These exercises in PDF To other course material