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Which spaces $X$ have the property that $X \to Y$ is continuous if and only if $I^n \to X \to Y$ is continuous for all $I^n \to X$?

Example: manifolds and CW complexes have this property, since we have surjective maps $I^n \to \Delta^n$ and $\bigcup_{k \geq 0} I^n \to \bigcup_{k \geq 0} \mathrm{Ball}(0, k) = \mathbb{R}^n$. This concludes since we can always deduce continuity “locally” on CW complexes and manifolds.

I am particularly interested to the case $X=\mathbb{E}_{\infty}(n)$ for all $n$. This would settle a minor doubt I had on $\mathbb{E}_{\infty}$-ring spectra, but I think the context is irrelevant.

Andrea Marino
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  • Is $n$ a fixed natural? – Jakobian Dec 02 '22 at 11:30
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    These are the numerically generated spaces. – Tyrone Dec 02 '22 at 13:29
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    Near-duplicate: https://math.stackexchange.com/questions/1406162/do-there-exist-general-conditions-under-which-we-can-conclude-that-continuity-on (It is easy to see that the condition there is equivalent to yours) – Eric Wofsey Dec 02 '22 at 20:16
  • Nice! It is enough for me to know that $\mathbb{E}_{\infty}(n)$ Is $\Delta$-generated for all $n$. I will add this example to the question. If you mind to answer to the example (follows from cocontinuity and the observation that $Conf_n(\mathbb{R}^k) $ is a manifold), I will accept your answer. – Andrea Marino Dec 03 '22 at 14:00

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