考虑配备标准欧几里得距离的度量空间\(\mathbb{R}^2\)
\ [d \大(x_1、x_2), (y_1 y_2) \大)= \√6 {(x_1 - y_1) ^ 2 + (x_2 - y_2) ^ 2}。\]
下面有多少个子集(S \子集\mathbb{R}^2\)在这个度量空间中是闭合的?
- \(S = \{(x,y) \,: \, x²+y²= 1\}\)
- \(S = \{(x,y) \,: \, x²+y²\le 1\}\)
- \ (S = \ {(x, y ) \, : \, x \ \ mathbb {Q}, y \ \ mathbb {Q} \} \)
- \ (S = \ {(x, 0 ) \, : \, C x \ \ mathcal{} \} \),在那里\ (\ mathcal C{} \ \子集mathbb {R} \)是middle-thirds康托尔集