Consider ℝ with the usual topology. Which of the following assertions is correct?

Explanation -

For option (1):

Any Hausdorff topology on a finite set is discrete. So it cannot have 3 different topologies.

For option (2):

Since X is finite and Hausdorff. So X × X is also finite and Hausdorff. So X × X has discrete topology. Hence every function f : X × X → ℝ is continuous.

For option (3):

Since f: ℝ → (X, discrete) is continuous.

So f is constant map and hence gof has any one element.

For option (4):

Suppose A is finite and dense i.e. A̅ = X. Now since A is finite set in a metric space.

⇒ A is closed set i.e. A̅ = A

⇒A = X so X is finite metric space and hence topology on X is discrete. So every f : X → ℝ is continuous.

Hence option (2) is true.