Pick the correct statement(s) from below. (A) $X = \prod _ { n = 1 } ^ { \infty } X _ { n }$ where $X _ { n } = \left\{ 1,2 , \ldots , 2 ^ { n } \right\}$ for $n \geq 1$ is not compact in the product topology. (B) $Y = \prod _ { n = 1 } ^ { \infty } Y _ { n }$ where $Y _ { n } = \left[ 0,2 ^ { n } \right] \subseteq \mathbb { R }$ for $n \geq 1$ is path-connected in the product topology. (C) $Z = \prod _ { n = 1 } ^ { \infty } Z _ { n }$ where $Z _ { n } = \left( 0 , \frac { 1 } { n } \right) \subseteq \mathbb { R }$ for $n \geq 1$ is compact in the product topology. (D) $P = \prod _ { n = 1 } ^ { \infty } P _ { n }$ where $P _ { n } = \{ 0,1 \}$ for $n \geq 1$ (with product topology) is homeomorphic to $( 0,1 )$.
Pick the correct statement(s) from below.\\
(A) $X = \prod _ { n = 1 } ^ { \infty } X _ { n }$ where $X _ { n } = \left\{ 1,2 , \ldots , 2 ^ { n } \right\}$ for $n \geq 1$ is not compact in the product topology.\\
(B) $Y = \prod _ { n = 1 } ^ { \infty } Y _ { n }$ where $Y _ { n } = \left[ 0,2 ^ { n } \right] \subseteq \mathbb { R }$ for $n \geq 1$ is path-connected in the product topology.\\
(C) $Z = \prod _ { n = 1 } ^ { \infty } Z _ { n }$ where $Z _ { n } = \left( 0 , \frac { 1 } { n } \right) \subseteq \mathbb { R }$ for $n \geq 1$ is compact in the product topology.\\
(D) $P = \prod _ { n = 1 } ^ { \infty } P _ { n }$ where $P _ { n } = \{ 0,1 \}$ for $n \geq 1$ (with product topology) is homeomorphic to $( 0,1 )$.