Consider a compact, Hausdorff topological space which is homeomorphic to its own copower over an index set $I$, so $X \cong \prod_{i \in I } X$. Is there necessarily another topological space, which is not a copower of any other space, whose copower over an index set is $X$? In other words, if a space is homeomorphic to its own copower, then can it be decomposed to some sort of atomic space?
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2$\begingroup$ Some Googling turns up a few uses of 'copower', but, since you mention a repeated selfproduct, you seem to be using it synonymously with 'power'. What is the difference? $\endgroup$ – LSpice Nov 14 '14 at 19:56

1$\begingroup$ The terminology can be confusing: this is actually a power object, sometimes called a cotensor object. The dual (repeated disjoint union) is a copower object or tensor object. $\endgroup$ – Tim Campion Nov 14 '14 at 22:30