Correct answer: (B) P is bounded.; (C) P is closed.
Explanation
All norms equal 1, proving boundedness. If a sequence of distinct exponents tends to infinity, its pointwise limit is zero on [0,1) and one at 1, hence it cannot converge uniformly to a continuous function. Any convergent sequence from P must therefore have bounded exponents and its limit belongs to P: P is closed. The subsequence x^(2^k) is pairwise separated by at least 1/4 (maximize t−t²), refuting total boundedness. Small constant perturbations leave P, refuting openness. **B,C**.
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