List of top Topology Questions on Compactness asked in GATE MA

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?

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 the following statements:
P: Every compact metrizable topological space is separable.
Q: Every Hausdorff topology on a finite set is metrizable.
Then,