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GATE 2024 MA – Question 50

Topology · Countability and Separation · 2 marks · Multiple select

The intervals [a,b),a<b form a topology basis on ℝ. Which statements hold?

  1. Every ordinary interval (a,b) is open.
  2. Every [a,b],a<b is compact.
  3. The space is first countable.
  4. The space is second countable.

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Show answer and explanation

Correct answer: (A) Every ordinary interval (a,b) is open.; (C) The space is first countable.

Explanation

Ordinary intervals are unions of basis intervals, proving A. At x, [x,x+1/n) is a countable local basis, proving C. The cover {[x,b):a≤x<b} together with [b,b+1) has no finite subcover of [a,b], so B fails. Any basis must supply, for each x, a member containing x and contained in [x,x+1); its least point is x, requiring uncountably many distinct basis elements. D fails. **A,C**.

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