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Part CCSIR NET December 2025compact-hausdorff-and-connected-survive-any-product-size-metrizable-needs-a-countable-index-set

Compact hausdorff and connected survive any product size metrizable needs a countable index set

Consider X=∏c∈(0,∞)X = \prod_{c\in(0,\infty)} [0,c] together with the product topology, where [0,c] is equipped with the euclidean topology. Which of the following statements are necessarily true?

  1. A.X is metrizable.
  2. B.X is compact.✓
  3. C.X is Hausdorff.✓
  4. D.X is connected.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Related counterexample: Countably compact implies compact

More on this topic

The chapter behind this: Compactness in general spaces — free to read

From Topology › Compactness and Tychonoff

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