Week 7 Exercises
Version 2024/11/20. These exercises in PDF To other course material
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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:
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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 \(\RR .\)
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1. Is \(\mathbb Q\) Hausdorff? Is \(\mathbb Q\) compact? Justify your answer.
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2. Show that the topology on \(\mathbb Q\) is not discrete.
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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.
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Version 2024/11/20. These exercises in PDF To other course material