List of top Topology Questions on Topology asked in GATE MA

Consider the following topologies on the set \( \mathbb{R} \) of all real numbers. \[ T_1 \text{ is the upper limit topology having all sets } (a, b) \text{ as basis.} \] \[ T_2 = \{ U \subset \mathbb{R}: U \text{ is finite} \} \cup \{\emptyset\}. \] \[ T_3 \text{ is the standard topology having all sets } (a, b) \text{ as basis.} \] Then:

Let \( \mathbb{R} \) denote the set of all real numbers. Consider the following topological spaces. \[ X_1 = (\mathbb{R}, T_1), \text{where} \, T_1 \text{ is the upper limit topology having all sets } (a, b) \text{ as basis.} \] \[ X_2 = (\mathbb{R}, T_2), \text{where} \, T_2 = \{ U \subset \mathbb{R} : \mathbb{R} \setminus U \text{ is finite} \} \cup \{\emptyset\}. \] Then: