List of top Topology Questions 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?

Let \( X = \{5, 6, 7, 8, 9, 10\} \) be equipped with the topology
\[ \tau = \{ \phi, X, \{5,6,7\}, \{8,9,10\} \} \]
Then the number of subsets of \( X \) which are neither open nor closed is ______.

Let \(X\) and \(Y\) be topological spaces and \(f: X \to Y\) be a continuous and bijective mapping. Which one of the following statements is TRUE?

The number of distinct topologies on the set \(\{1, 2, 3\}\) consisting of exactly four elements is equal to ______. (Answer in integer)

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?

Let \( X \) be a set with at least two elements. Let \( d_1, d_2 \) and \( d_3 \) be metrics on \( X \).
Which one of the following is NOT a metric on \( X \)?