List of top Topology Questions on Basic concepts of topology asked in GATE MA

Consider \( ([0, 1], T_1) \), where \( T_1 \) is the subspace topology induced by the Euclidean topology on \( \mathbb{R} \), and let \( T_2 \) be any topology on \( [0, 1] \). Consider the following statements: P: If \( T_1 \) is a proper subset of \( T_2 \), then \( ([0, 1], T_2) \) is not compact. Q: If \( T_2 \) is a proper subset of \( T_1 \), then \( ([0, 1], T_2) \) is not Hausdorff. Then, which of the following statements is TRUE?

Consider \( \mathbb{R}^2 \) with the usual metric. Let \[ A = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\} \quad \text{and} \quad B = \{(x, y) \in \mathbb{R}^2 : (x - 2)^2 + y^2 \leq 1\}. \] Let \( M = A \cup B \) and \( N = \text{interior}(A) \cup \text{interior}(B) \). Then, which of the following statements is TRUE?

Let \( (\mathbb{Q}, d) \) be the metric space with \[ d(x, y) = |x - y|. \] Let \( E = \{ p \in \mathbb{Q} : 2<p^2<3 \}. \) Then, the set \( E \) is

Consider the following topologies on the set \( \mathbb{R} \) of all real numbers:
\( T_1 = \{ U \subset \mathbb{R} : 0 \notin U \text{ or } U = \mathbb{R} \} \),
\( T_2 = \{ U \subset \mathbb{R} : 0 \in U \text{ or } U = \emptyset \} \),
\( T_3 = T_1 \cap T_2 \).
Then the closure of the set \( \{1\} \) in \( (\mathbb{R}, T_3) \) is