Saaqib Mahmood

### Questions (720)

 17 How to prove that if a normed space has Schauder basis, then it is separable? What about the converse? 17 How to prove the continuity of the metric function? 14 Prob. 17, Chap. 2 in Baby Rudin: The set of all numbers in $[0,1]$ with only $4$ and $7$ as decimal digits is countable, dense, compact, perfect? 13 Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof 12 Prob. 2(b), Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The iff-condition for two points to be in the same component of $\mathbb{R}^\omega$

### Reputation (8,599)

 +5 Prob. 21, Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets +5 No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof? +5 How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$? +5 Is the space $B([a,b])$ separable?

### Answers (99)

 8 Construct a bijection from $(0,1]$ to $(0,1)$ and $[0,1]$ to $(0,1)$ 6 Baby Rudin theorem 2.41: need a rigorous construction when $E$ is not closed. 4 Combination of all kinds of basic series stuffed together. 4 absolute convergence implies unconditional convergence 3 Finding number of solutions in the positive integers of (i) $x_1x_2x_3x_4x_5 = 1260$ (ii) $2x + 3y + 4z = 24$

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 21 real-analysis × 371 7 combinatorics × 10 11 calculus × 128 5 analysis × 392 9 elementary-number-theory × 16 5 linear-algebra × 32 8 real-numbers × 4 4 general-topology × 202 7 sequences-and-series × 102 4 functional-analysis × 123

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