Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Topology Separation Axioms
Consider ℝ with the usual topology. Which of the following assertions is correct?
1
A finite set containing 33 elements has at least 3 different Hausdorff topologies.
2
Let X be a non-empty finite set with a Hausdorff topology. Consider X × X with the product topology. Then, every function f ∶ X × X → ℝ is continuous.
3
Let X be a discrete topological space having infinitely many elements. Let f ∶ ℝ → X be a continuous function and g ∶ X → ℝ be any non-constant function. Then the range of g ∘ f contains at least 2 elements.
4
If a non-empty metric space X has a finite dense subset, then there exists a discontinuous function f ∶ X → ℝ.