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by Stuart Armstrong 256 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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