Define the terms connected and path-connected for a topological space. Prove that the interval $[0,1]$ is connected and that if a topological space is path-connected, then it is connected.

Let $X$ be an open subset of Euclidean space $\mathbb{R}^{n}$. Show that $X$ is connected if and only if $X$ is path-connected.

Let $X$ be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in $\mathbb{R}^{n}$. Assume $X$ is connected; must $X$ be also pathconnected? Briefly justify your answer.

Consider the following subsets of $\mathbb{R}^{2}$ :

$$\begin{gathered} A=\{(x, 0): x \in(0,1]\}, \quad B=\{(0, y): y \in[1 / 2,1]\}, \text { and } \\ C_{n}=\{(1 / n, y): y \in[0,1]\} \text { for } n \geqslant 1 \end{gathered}$$

Let

$$X=A \cup B \cup \bigcup_{n \geqslant 1} C_{n}$$

with the subspace topology. Is $X$ path-connected? Is $X$ connected? Justify your answers.