It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval. (A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$. (B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.
It is known that there exist surjective continuous maps $I \longrightarrow I ^ { 2 }$ where $I = [ 0,1 ]$ is the unit interval.\\
(A) (4 marks) Using the above result or otherwise, show that there exists a surjective continuous map $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$.\\
(B) (6 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R } ^ { 2 }$ be a surjective continuous map. Let $\Gamma = \{ ( x , f ( x ) ) \mid x \in \mathbb { R } \} \subset \mathbb { R } ^ { 3 }$. Show that $\mathbb { R } ^ { 3 } \setminus \Gamma$ is path connected.