Consider the following subsets of $\mathbb { R } ^ { 2 } : X _ { 1 } = \left\{ \left. \left( x , \sin \frac { 1 } { x } \right) \right\rvert \, 0 < x < 1 \right\} , X _ { 2 } = [ 0,1 ] \times \{ 0 \}$, and $X _ { 3 } = \{ ( 0,1 ) \}$. Then, (a) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a connected set; (b) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a path-connected set; (c) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is not path-connected, but $X _ { 1 } \cup X _ { 2 }$ is path-connected; (d) $X _ { 1 } \cup X _ { 2 }$ is not path-connected, but every open neighbourhood of a point in this set contains a smaller open neighbourhood which is path-connected.
Consider the following subsets of $\mathbb { R } ^ { 2 } : X _ { 1 } = \left\{ \left. \left( x , \sin \frac { 1 } { x } \right) \right\rvert \, 0 < x < 1 \right\} , X _ { 2 } = [ 0,1 ] \times \{ 0 \}$, and $X _ { 3 } = \{ ( 0,1 ) \}$. Then,\\
(a) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a connected set;\\
(b) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is a path-connected set;\\
(c) $X _ { 1 } \cup X _ { 2 } \cup X _ { 3 }$ is not path-connected, but $X _ { 1 } \cup X _ { 2 }$ is path-connected;\\
(d) $X _ { 1 } \cup X _ { 2 }$ is not path-connected, but every open neighbourhood of a point in this set contains a smaller open neighbourhood which is path-connected.