Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'': $$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$ Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below. (A) $Y$ is compact. (B) $X$ and $Y$ are connected. (C) $Z$ is not compact. (D) $Z$ is path-connected.
Let $X$ be the metric space of real-valued continuous functions on the interval $[ 0,1 ]$ with the ``supremum distance'':
$$d ( f , g ) = \sup \{ | f ( x ) - g ( x ) | : x \in [ 0,1 ] \} \text { for all } f , g \in X$$
Let $Y = \{ f \in X : f ( [ 0,1 ] ) \subset [ 0,1 ] \}$ and $Z = \left\{ f \in X : f ( [ 0,1 ] ) \subset \left[ 0 , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , 1 \right] \right\}$. Pick the correct statement(s) from below.\\
(A) $Y$ is compact.\\
(B) $X$ and $Y$ are connected.\\
(C) $Z$ is not compact.\\
(D) $Z$ is path-connected.