1. Please explain “NP-completeness (NPC)” and give several NPC problems. (25%)

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?