Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Topology Topological spaces and Continuous Functions
Let X be a set and T1 and T2 be two distinct topologies on X. Consider the set T = {T1, T2} and the usual subset topology on 2X (the power set of X). Which of the following is correct?
1
\((2^X, T)\) is always a topological space.
2
\((2^X, T)\) is never a topological space
3
\((2^X, T)\) is a topological space if and only if T1 is a subset of T2 or T2 is a subset of T1.
4
The property of \((2^X, T)\) being a topological space depends on the specific sets T1 and T2, but cannot be determined merely from set inclusion relations between the two.