Week 8 Exercises
Version 2024/11/20. These exercises in PDF To other course material
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Exercise 8.0. This is an unseen exercise on closure, boundary and dense sets. Consider the sets \(A=\{0,1\}\subset \RR \) and \(B=\RR \setminus A = (-\infty , 0)\cup (0,1) \cup (1,+\infty )\) as a subsets of four different topological spaces, given in the table below. Complete the table.
The space \(X\) \((\RR ,\) antidiscrete\()\) \((\RR ,\) cofinite\()\) \((\RR ,\) Euclidean\()\) \((\RR ,\) discrete\()\) \(\overline A\) (closure in \(X\)) Is \(A\) dense in \(X\)? (yes/no) \(\overline B\) Is \(B\) dense in \(X\)? (yes/no) \(\partial A\) Hint. You may wish to recall that \(\overline A = \) the smallest closed set in \(X\) which contains \(A\) = \(\{z\in X:\) all open neighbourhoods of \(z\) meet \(A\}\) and that \(\partial A = \overline A \cap \overline {(X\setminus A)}.\)
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Exercise 8.1. (a) Use the following two results,
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• a connected component of a topological space is a connected set (Lemma 7.4),
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• if the space \(X\) has a connected dense subset then \(X\) is connected (Lemma 7.11),
to show that each connected component of a topological space is a closed set.
(b) Deduce from (a) that if a topological space \(X\) has finitely many connected components, then each connected component is both closed and open in \(X.\)
(c) Give an example of a topological space where connected components are closed but not open.
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Exercise 8.2. (a) Suppose that \(X\) is a topological space, points \(x,y\in X\) are joined by a path in \(X,\) and points \(y,z\in X\) are also joined by a path in \(X.\) Show that \(x,z\) are joined by a path in \(X.\)
(b) Furthermore, show that “\(x\sim y\) \(\iff \) \(x,y\) are joined by a path in \(X\)” is an equivalence relation on \(X.\)
Equivalence classes defined by the relation \(\sim \) from (b) are called path-connected components of \(X.\) In general, a path-connected component does not need to be open or closed in \(X.\) Nevertheless:
(c) Show that if \(X\) is an open subset of a Euclidean space \(\RR ^n,\) then each path-connected component of \(X\) is open. Deduce that an open connected subset of \(\RR ^n\) is path-connected.
Version 2024/11/20. These exercises in PDF To other course material