by Stuart Armstrong 568 days ago | Scott Garrabrant likes this | link | parent Can you argue that $$X$$ must have a semi-metric compatible with the topology by using $$d(x,y)=sup_{z\in X}|h(x,z)-h(y,z)|$$? I’m wondering if you can generalise this to some sort of argument that goes like this. Using X, project down via $$\pi$$ from $$X^0=X$$ to $$X^1=X^0/d$$. Let $$\phi$$ be our initial surjection; it’s now a bijection between $$X^1$$ and maps from $$X^0$$ to $$[0,1]$$. If the projection is continuous, then every map from $$X^1$$ to $$[0,1]$$ lifts to a map from $$X^0$$ to $$[0,1]$$. Restricting to the subset of maps that are lifts like this, and applying $$\phi^{-1}$$, gives a subset $$X^2 \subset X^1$$. We now have a new equivalence relationship, maps from $$X^1$$ that are equal to each other on $$X^2$$. Project down from $$X^2$$ by this relationship, to generate $$X^3$$. Continue this transfinitely often (?) to generate a space $$X'$$ where $$\phi$$ is a homeomorphism, and find a contradiction? This feels dubious, but maybe worth mentioning…

 by Alex Mennen 567 days ago | Scott Garrabrant and Stuart Armstrong like this | link I haven’t checked that argument carefully, but that sounds like it should give you $$X'$$ with a continuous bijection $$\phi:X'\rightarrow[0,1]^{X'}$$, which might not necessarily be a homeomorphism. reply
 by Stuart Armstrong 567 days ago | link Yes, you’re right. reply

### NEW DISCUSSION POSTS

[Note: This comment is three
 by Ryan Carey on A brief note on factoring out certain variables | 0 likes

There should be a chat icon
 by Alex Mennen on Meta: IAFF vs LessWrong | 0 likes

Apparently "You must be
 by Jessica Taylor on Meta: IAFF vs LessWrong | 1 like

There is a replacement for
 by Alex Mennen on Meta: IAFF vs LessWrong | 1 like

Regarding the physical
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think that we should expect
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think I understand your
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

This seems like a hack. The
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

After thinking some more,
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yes, I think that we're
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

My intuition is that it must
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

To first approximation, a
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Actually, I *am* including
 by Vadim Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yeah, when I went back and
 by Alex Appel on Optimal and Causal Counterfactual Worlds | 0 likes

> Well, we could give up on
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes