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.

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 \(\partial A = \overline A \cap \overline {(X\setminus A)}.\)

Exercise 8.1. (a) Use the following two results,

• • a connected component of a topological space is a connected set (Lemma 7.4),

• • 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.

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.