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Is this true?

A compact space is Hausdorff

No — it is false.

The counterexample

An infinite set with the cofinite topology

Compact and but any two non-empty open sets intersect.

The kind of mistake this is

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this
topology

Others that fail the same way

From Topology › Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

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