The GATE Grind

GATE 2022 MA – Question 53

Topology · Countability and Separation · 2 marks · Multiple select

Let T be the smallest topology on ℝ for which every singleton is closed. For X=(ℝ,T), which hold?

  1. [0,1) is compact.
  2. X is not first countable.
  3. X is second countable.
  4. X is first countable.

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Show answer and explanation

Correct answer: (A) [0,1) is compact.; (B) X is not first countable.

Explanation

T is the cofinite topology. Any subset is compact: pick one nonempty covering open set, which leaves only finitely many points to cover. If (U_n) were a countable local base at x, the union of finite complements would be countable. Choose y≠x outside that union. Every U_n contains y, so none lies in the neighborhood ℝ∖{y}, a contradiction. Thus X is not first countable and therefore not second countable. **A,B**.

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