List of top Topology Questions on Compactness and Connectedness asked in GATE MA

Let \(S = \bigcup_{n=1}^{\infty} \left\{ (x, y) \in \mathbb{R}^2 : (x - n)^2 + y^2 = \dfrac{1}{n^2} \right\}\) with the usual topology. Which of the following statements is/are TRUE?

Consider \(\mathbb{R}\) with the topology
\[ \tau = \{A \subseteq \mathbb{R} : A^c \text{ is finite}\} \cup \{\phi\}. \]
Which of the following statements is/are TRUE?