For P, choose a finite 1/n-net for each n using compactness. Their countable union is dense. For Q, finite Hausdorff spaces have closed singletons and thus every set is closed/open; the topology is discrete and is induced by the discrete metric. Both true, **A**.
More Countability and Separation questions
GATE 2026 MA Q47– A map f:X→Y is continuous and bijective. Which condition guarantees it is a homeomorphism?
GATE 2025 MA Q58– X is uncountable with cocountable topology: ∅ is open and every set with finite or…
GATE 2024 MA Q17– I: Every compact Hausdorff space is normal. II: Every metric space is normal. Choose truth…
GATE 2024 MA Q50– The intervals [a,b),a<b form a topology basis on ℝ. Which statements hold?
GATE 2023 MA Q44– On ℝ let τ=U⊆ℝ:U=∅ or 1∈U. Which statements hold?
GATE 2022 MA Q53– Let T be the smallest topology on ℝ for which every singleton is closed. For X=(ℝ,T),…