2 (15) A set \( A \subseteq \mathbb{R}^n \) is said to be dense in \( B \subseteq \mathbb{R}^n \) if \( B \subseteq \operatorname{cl}(A) \). If \( A \) is dense in \( \mathbb{R}^n \) and \( U \) is open, prove that \( A \cap U \) is dense in \( U \). Is this true if \( U \) is not open?

阿摩線上測驗 · 2026-11-13T14:01:06+08:00

2 (15) A set A R^n is said to be dense in B R^n if B cl(A). If A is dense in R^n and U is open, prove that A U is dense in U. Is this true if U is not open?

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2 (15) A set \( A \subseteq \mathbb{R}^n \) is said to be dense in \( B \subseteq \mathbb{R}^n \) if \( B \subseteq \operatorname{cl}(A) \). If \( A \) is dense in \( \mathbb{R}^n \) and \( U \) is open, prove that \( A \cap U \) is dense in \( U \). Is this true if \( U \) is not open?

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