Mathematics · Topology
Consider ℝ with the usual topology. Which of the following assertions is correct?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If a non-empty metric space X has a finite dense subset, then there exists a discontinuous function f : X → ℝ.
- 2Let 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.
- 3Let 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.✓
- 4A finite set containing 33 elements has at least 3 different Hausdorff topologies.
Correct answer
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.
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