If compactness depends on topology, what are good examples for "alternative" topologies of spaces which steal away their compactness, i.e. I ask for spaces which are usually considered together with a topology such that they are compact and then topologies in which they are not.
Example for changing the compactness of a manifold by considering another topology
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1I think a good example occurs in functional analysis: in the dual space of an infinite-dimensional Banach space, the closed unit ball is not compact in the (usual) operator norm topology, but when you give that unit ball the weak-star topology it becomes compact. This is the Banach-Alaoglu theorem. – 2012-06-07
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Any topology on a finite set is compact, so you won't find any examples there.
However, given any infinite topological space, replacing the topology with the discrete topology always makes it non-compact.
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1Sure, but this doesn't actually have any applications. – 2012-06-07