List of top Topology Questions

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?
Let \( p : ([0, 1], T_1) \to \{(0, 1\}, T_2) \) be the quotient map, arising from the characteristic function on \( [\frac{1}{2}, 1] \), where \( T_1 \) is the subspace topology induced by the Euclidean topology on \( \mathbb{R} \). Which of the following statements is TRUE?
Set \( X_n := \mathbb{R} \) for each \( n \in \mathbb{N} \). Define \( Y := \prod_{n \in \mathbb{N}} X_n \). Endow \( Y \) with the product topology, where the topology on each \( X_n \) is the Euclidean topology. Consider the set \[ \Delta = \{ (x, x, x, \dots) \mid x \in \mathbb{R} \} \] with the subspace topology induced from \( Y \). 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 \( X = (\mathbb{R}, T) \), where \( T \) is the smallest topology on \( \mathbb{R} \) in which all the singleton sets are closed. Then, which of the following statements are TRUE?
Consider \( (\mathbb{Z}, T) \), where \( T \) is the topology generated by sets of the form \[ A_{m,n} = \{ m + nk \mid k \in \mathbb{Z} \}, \quad \text{for } m, n \in \mathbb{Z} \text{ and } n \neq 0. \] Then, which of the following statements are 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