Department of Mathematics, K.T.H.M. College, Nashik

mathsbeauty

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Registered: 11-06-2015

Posts: 14

Compact subsets (Part B Question: NET June-2015)

(Part B Question: NET June-2015)Which of the following subsets of \(\mathbb{R}^n \) is compact (w.r.t. usual topology of \(\mathbb{R}^n)\) ?
A. \(\{(x_1,x_2,\ldots,x_n):|x_i|<1,1\leq i\leq n\} \)
B. \(\{(x_1,x_2,\ldots,x_n):x_1+x_2+\cdots+x_n = 0\} \)
C. \(\{(x_1,x_2,\ldots,x_n):x_i\geq 0,1\leq i\leq n\} \)
D. \(\{(x_1,x_2,\ldots,x_n):1\leq i\leq 2^i, 1 \leq i \leq n\} \)

\(\textbf{Option D}\)
The set  \(\{(x_1,x_2,\ldots,x_n):|x_i|<1,1\leq i\leq n\} \) is not closed.
The set \(\{(x_1,x_2,\ldots,x_n):x_1+x_2+\cdots+x_n = 0\} \) is not bounded.
The set \(\{(x_1,x_2,\ldots,x_n):x_i\geq 0,1\leq i\leq n\} \) is not bounded.
The set \(\{(x_1,x_2,\ldots,x_n):1\leq i\leq 2^i, 1 \leq i \leq n\} \) is closed and bounded.
(\(\textbf{Hint:} \) Draw sets in \(\mathbb{R}^2) \)