Exercise 8.0. This is an unseen exercise on closure, boundary and dense sets. Consider the sets $A=\{0,1\}\subset \mathbb{R}$ and $B=\mathbb{R}\setminus A=(-\mathrm{\infty },0)\cup (0,1)\cup (1,+\mathrm{\infty })$ as a subsets of four different topological spaces, given in the table below. Complete the table.

Hint. You may wish to recall that $\bar{A}=$ the smallest closed set in $X$ which contains $A$ = $\{z\in X:$ all open neighbourhoods of $z$ meet $A\}$ and that $\partial A=\bar{A}\cap \overline{(X\setminus A)}.$

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 “$x\sim y$ $⟺$ $x,y$ 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 ${\mathbb{R}}^n,$ then each path-connected component of $X$ is open. Deduce that an open connected subset of ${\mathbb{R}}^n$ is path-connected.