by Stuart Armstrong 567 days ago | Scott Garrabrant likes this | link | parent A small note: it’s not hard to construct spaces that are a bit too big, or a bit too small (raising the possibility that a true $$X$$ lies between them). For instance, if $$I$$ is the unit interval, then we can map $$I$$ onto the countable-dimensions hypercube $$I^\omega$$ ( https://en.wikipedia.org/wiki/Space-filling_curve#The_Hahn.E2.80.93Mazurkiewicz_theorem ). Then if we pick an ordering of the dimensions of the hypercube and an ordering of $$\mathbb{Q}\cap I$$, we can see any element of $$I^\omega$$ - hence any element of $$I$$ - as a function from $$\mathbb{Q}\cap I$$ to $$I$$. Let $$C(I)$$ be the space of continuous functions $$I \to I$$. Then any element of $$C(I)$$ defines a unique function $$\mathbb{Q}\cap I \to I$$ (the converse is not true - most functions $$\mathbb{Q}\cap I \to I$$ do not correspond to continuous functions $$I \to I$$). Pulling $$C(I)$$ back to $$I$$ via $$I^\omega$$ we define the set $$Y \subset I$$. Thus $$Y$$ maps surjectively onto $$C(I)$$. However, though $$C(I)$$ maps into $$C(Y)$$ by restriction (any function from $$I$$ is a function from $$Y$$), this map is not onto (for example, there are more continuous functions from $$I - \{1/2\}$$ than there are from $$I$$, because of the potential discontinuity at $$1/2$$). Now, there are elements of $$I-Y$$ that map (via $$I^\omega$$) to functions in $$C(Y)$$ that are not in $$C(I)$$. So there’s a hope that there may exist an $$X$$ with $$Y \subset X \subset I$$, $$C(I) \subset C(X) \subset C(Y)$$, and $$X$$ mapping onto $$C(X)$$. Basically, as $$X$$ `gets bigger’, its image in $$C(Y)$$ grows, while $$C(X)$$ itself shrinks, and hopefully they’ll meet.

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