Timeline for Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation
Current License: CC BY-SA 3.0
12 events
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| Mar 6, 2012 at 13:56 | vote | accept | Martin Brandenburg | ||
| Mar 6, 2012 at 13:56 | comment | added | Martin Brandenburg | Hehe, indeed! I like this example. | |
| Mar 4, 2012 at 23:29 | comment | added | Ralph | Is this the confirmation that the example satisfies "some computation has to be done to find the point $x\in X$ such that $F$ is the evaluation at $x$ ?" -:) | |
| Mar 4, 2012 at 23:15 | history | edited | Ralph | CC BY-SA 3.0 |
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| Mar 3, 2012 at 21:17 | comment | added | Martin Brandenburg | @Ralph: Thanks for the update! Do we have $F(f)=f(\omega_1)$ in your example? | |
| Mar 3, 2012 at 17:16 | history | edited | Ralph | CC BY-SA 3.0 |
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| Mar 3, 2012 at 17:10 | history | edited | Ralph | CC BY-SA 3.0 |
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| Mar 3, 2012 at 16:00 | comment | added | Martin Brandenburg | Ok, but isn't this just a fancy way to state the old result that prime ideals of a product of fields, indexed by a set $N$, correspond to ultrafilters on $N$? | |
| Mar 3, 2012 at 12:42 | comment | added | Gerald Edgar | As Ralph hints, when the space $X$ is not a Q-space (or, in other language, not realcompact) there is a countably-additive example of a zero-one Baire measure that is not fixed. The set of such things constitutes $\upsilon X$, the Hewitt realcompactification of $X$. | |
| Mar 3, 2012 at 12:39 | comment | added | Gerald Edgar | Space $\mathbb N$, the zero-one finitely additive measure is an ultrafilter. They define homomorphisms, and the "points" corresponding to them are the points of the Stone-Cech compactification $\beta \mathbb N$. So: it is not an evaluation to start with, but then we "invent" the points to make it so. | |
| Mar 3, 2012 at 8:49 | comment | added | Martin Brandenburg | Can you give some specific example, especially where $\mu$ isn't a dirac measure? | |
| Mar 3, 2012 at 1:17 | history | answered | Ralph | CC BY-SA 3.0 |