For the (Hausdorff) compact spaces I can think of, compactness is established either using a product of compact spaces (including the Heine-Borel Theorem, the Banach-Alaoglu Theorem, Stone-Čech compactification, etc.) or by inheriting compactness from another space (e.g. the Hausdorff metric on compact subsets of another compact space). I guess it might be vague as to whether one-point/end type compactifications fall into the second category, but compactness is generally established from compact subsets of the original space.
Are there any examples of compact spaces whose compactness can be established via a fundamentally different method?