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It is known that no two distinct finite powers of the closed unit interval are homeomorphic:

$I^m$ is homeomorphic to $I^n$ iff $m=n$. (Brouwer, Lebesgue, 1911)

Is the analogous result for infinite powers of $I$ true?

That is, is it true that $I^\alpha$ is homeomorphic to $I^\beta$ iff $\alpha=\beta$, for cardinal numbers $\alpha$ and $\beta$?

(An affirmative answer would give a hope of defining a transfinite inductive dimension for all spaces, a problem essentially posed by Carl Menger.)

If not, what are the homeomorphism classes among the infinite powers of $I$?

Added Later: Since $|I^\alpha|=2^{\aleph _0 \alpha}=2^\alpha$ whenever $\alpha$ is infinite, assuming the Generalised Continuum Hypothesis, of course, $2^\alpha=2^\beta \Rightarrow \alpha=\beta$ and $I^\alpha$ and $I^\beta$ have different cardinalities if $\alpha$ and $\beta$ are distinct. But the answer below shows that they can be proved to be not homeomorphic within ZFC, and, more interestingly, even when they have the same cardinality (in some model which violates GCH).

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1 Answer 1

up vote 31 down vote accepted

Yes. $I^{\alpha}=I^{\beta}$ does imply that $\alpha=\beta$. To see this, suppose that $\alpha$ is an infinite cardinal. Then each point in $I^{\alpha}$ is the intersection of $\alpha$ many open sets, but each point in $I^{\alpha}$ is not the intersection of less than $\alpha$ many open sets.

In greater detail, if $(x_{\beta})_{\beta<\alpha}\in I^{\alpha}$, then for each $n\in\mathbb{N}$ and $J\subseteq\alpha$ with $|J|<\aleph_{0}$, let $U_{J,n}=\{(y_{\beta})_{\beta<\alpha}:|y_{j}-x_{j}|<\frac{1}{n}\,\textrm{for}\,j\in J\}$. Then $(x_{\beta})_{\beta<\alpha}=\bigcap_{J,n}U_{J,n}$, so $(x_{\beta})_{\beta<\alpha}$ is the intersection of $\alpha$ many open sets.

Now, suppose $|J|<\alpha$ and $U_{i}\subseteq I^{\alpha}$ is an open neighborhood of $(x_{\beta})_{\beta<\alpha}$ for $i\in J$. Then for each $i\in J$ there is a finite $J_{i}\subseteq\alpha$ where if $(y_{\beta})_{\beta\in\alpha}\in I^{\alpha}$ and $y_{j}=x_{j}$ for $j\in J_{i}$, then $(y_{\beta})_{\beta\in\alpha}\in U_{i}$. Since $|\bigcup_{i\in J}J_{i}|=|J|<\alpha$, $\bigcup_{i\in J}J_{i}\neq\alpha$. Therefore, whenever $y_{\beta}=x_{\beta}$ for $\beta\in\bigcup_{i\in J}J_{i}$, we have $(y_{\beta})_{\beta\in \alpha}\in\bigcap_{i\in J}U_{i}$, but we do not necessarily have $(y_{\beta})_{\beta\in \alpha}=(x_{\beta})_{\beta<\alpha}$. Hence, $\bigcap_{i\in J}U_{i}$ is not equal to $\{(x_{\beta})_{\beta<\alpha}\}$.

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Great answer and fast enough too, though I was a bit confused by the use of $I$ in the last paragraph (and have suggested editing it to $J$). This answer also makes it clear exactly which cubes are perfectly normal etc., among other things, in an elegant way (I have not seen a direct proof in textbooks). Is there a reference? –  N Unnikrishnan Jun 3 at 17:40
    
I do not have a direct reference, but I simply found that $\alpha$ can be extracted from $I^{\alpha}$ by a cardinal invariant which for compact spaces is equivalent to the local character of a space. One should therefore look up cardinal invariants of a topological space for similar invariants that can be used to distinguish topological spaces. –  Joseph Van Name Jun 3 at 18:52
    
Can you specify that cardinal invariant? (I could not find one.) –  N Unnikrishnan Jun 3 at 22:15
5  
For an infinite cardinal $\alpha$, the cube $I^{\alpha}$ has many cardinal invariants equal to $\alpha$: weight, character, pseudo-character (as in Joseph´s answer), spread, tightness, and many more. You can look at Hodel´s article in the Handbook of Set Theoretic Topology for more information on those. –  Ramiro de la Vega Jun 5 at 15:35

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