Paper 1, Section II, E
What does it mean to say that a topological space is compact? Prove directly from the definition that is compact. Hence show that the unit circle is compact, proving any results that you use. [You may use without proof the continuity of standard functions.]
The set has a topology for which the closed sets are the empty set and the finite unions of vector subspaces. Let denote the set with the subspace topology induced by . By considering the subspace topology on , or otherwise, show that is compact.
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