Let $X$ be a subset of $\mathbb { R } ^ { 3 }$. We say that $X$ has property $S$ if it contains at least two elements and every Cauchy sequence in $X$ has a limit point in $X$. Pick the correct statement(s) from below. (A) If $X$ has property $S$ then it must be compact. (B) If $X$ has property $S$ then it must be closed. (C) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $\left( a _ { 1 } , b _ { 1 } , c _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } , c _ { 2 } \right) \in X$, then $\left( a _ { 1 } + a _ { 2 } , b _ { 1 } + b _ { 2 } , c _ { 1 } + c _ { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$. (D) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $( a , b , c ) \in X$, then $\left( \frac { a } { 2 } , \frac { b } { 2 } , \frac { c } { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.
Let $X$ be a subset of $\mathbb { R } ^ { 3 }$. We say that $X$ has property $S$ if it contains at least two elements and every Cauchy sequence in $X$ has a limit point in $X$. Pick the correct statement(s) from below.\\
(A) If $X$ has property $S$ then it must be compact.\\
(B) If $X$ has property $S$ then it must be closed.\\
(C) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $\left( a _ { 1 } , b _ { 1 } , c _ { 1 } \right) , \left( a _ { 2 } , b _ { 2 } , c _ { 2 } \right) \in X$, then $\left( a _ { 1 } + a _ { 2 } , b _ { 1 } + b _ { 2 } , c _ { 1 } + c _ { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.\\
(D) Suppose that $X$ has property $S$ and it further satisfies the following condition: if $( a , b , c ) \in X$, then $\left( \frac { a } { 2 } , \frac { b } { 2 } , \frac { c } { 2 } \right) \in X$. Then $X$ is dense in $\mathbb { R } ^ { 3 }$.