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Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ where each $Y_i \cong Y$. Are there any internal conditions on how $Y$ sits inside $X$ that guarantees it is a factor? Has any research been carried out as to when factors exist -- In other words, which spaces are products?

The only result of this type I know relates to continua. Since the product of two arbitrary continua is always aposyndetic, we can guarantee that any non-aposyndetic continuum does not have a factor. This relies heavily on connectedness though.

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  • $\begingroup$ In case of a Cantor set $X$ for the space $Y$ we can take any closed subset containing more than one point. If $X$ is the space of rationals, then for $Y$ we can take any non-compact $G_\delta$-subset of $Y$. This means that in case of zero-dimensional spaces we have a lot of freedom. $\endgroup$ – Taras Banakh Sep 22 '15 at 16:42

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