Timeline for Space of continuous real-valued functions on $[0,1]^\omega$ with the weak and pointwise topology
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2017 at 5:41 | comment | added | Taras Banakh | This problems was posed and studied (but not answered) by M.Krupski in his paper (arxiv.org/abs/1504.04554) posted to arXiv on 17 April 2015 (so three months earlier as Dominic asked this question here). Because of that the question should be attributed to M.Krupski. In his paper Krupski proved that for an infinite finite-dimensional compact metrizable space $K$ the function spaces $C_p(K)$ and $C_w(K)$ are not homeomorphic. In particular, they are not homeomorphic for $K=[0,1]^n$ where $n\in\mathbb N$. | |
| Jun 18, 2015 at 15:28 | comment | added | Eric Wofsey | For what it's worth, they are not isomorphic as topological vector spaces (for instance, the weak* topology on the dual with respect to $\sigma$ has the Heine-Borel property, but this is not true for $\tau$). But homeomorphism is a much weaker property than isomorphism of topological vector spaces (for instance, all separable infinite-dimensional Banach spaces are homeomorphic). | |
| Jun 18, 2015 at 15:01 | comment | added | Martin Sleziak | On the other hand, the example of net net which is convergent in one topology but not in the other one only shows that the identity map is not a homeomorphism. (Even if we are able to modify the argument from $C[0,1]$ to this case.) That still does not refute a possibility that there is some other homeomorphism. | |
| Jun 18, 2015 at 11:52 | comment | added | Dominic van der Zypen | Sorry -- I meant "homeomorphic" (as topological spaces) and not "isomorphic". Just edited the post accordingly | |
| Jun 18, 2015 at 11:51 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
isomorphic -> homeomorphic
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| Jun 18, 2015 at 11:38 | comment | added | Gerald Edgar | Isomorphic as vector spaces, yes. But not as topological vector spaces. Martin is correct: the method used for $C[0,1]$ also works for this. More generally, try proving it for $C(K)$, where $K$ is an infinite compact metric space. | |
| Jun 18, 2015 at 11:15 | comment | added | Eric Wofsey | Do you mean isomorphic as topological vector spaces? | |
| Jun 18, 2015 at 10:57 | comment | added | Martin Sleziak | This answer provides an example showing that pointwise convergence and weak convergence are not equivalent in $C([0,1])$. I suppose that the same example should work for product of countably many copies of $[0,1]$. (The answer was given by Theo Buehler.) | |
| Jun 18, 2015 at 10:40 | review | Close votes | |||
| Jun 18, 2015 at 11:15 | |||||
| Jun 18, 2015 at 10:40 | comment | added | Dominic van der Zypen | @MartinSleziak That's correct - thanks for asking! | |
| Jun 18, 2015 at 10:30 | comment | added | Martin Sleziak | By weak topology on $C(X)$ you mean the initial topology w.r.t. all functions from the dual $C^*(X)$ (i.e., all linear continuous functionals on $C(X)$)? | |
| Jun 18, 2015 at 9:46 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |