bio | website | andrej.com |
---|---|---|
location | Ljubljana | |
age | 43 | |
visits | member for | 5 years, 6 months |
seen | yesterday | |
stats | profile views | 7,410 |
I am a professional mathematician. My area of research is logic, constructive and computable mathematics, category theory, and semantics of programming languages.
Apr 18 |
revised |
Minimal totally separated spaces
editing just so I can cancel my downvote |
Apr 18 |
comment |
Minimal totally separated spaces
It does not matter. There are Hausdorff spaces which match your definition ("clopens separate"), but they are not $0$-dimensional in the standard sense (i.e., small inductive dimension). Is there a standard notion of dimension for which your definitions means "$0$-dimensional"? |
Apr 18 |
comment |
Minimal totally separated spaces
Ah, excellent. And so your last remark also shows that we do not even have to do anything with Stone spaces to find a counter-example: just take a totally disconnected space which is not $0$-dimensional. |
Apr 18 |
comment |
Minimal totally separated spaces
Oh I see, you're saying you'll find an example which is not only totally disconnected (what OP called "0d") but is also zero-dimensional (clopens form a basis). |
Apr 18 |
comment |
Minimal totally separated spaces
For small inductive dimension, zero-dimensionality is equivalent to "clopens form a basis" (see Wikipedia if you trust it, en.wikipedia.org/wiki/Zero-dimensional_space). A space is totally disconnected iff its components are singletons, see en.wikipedia.org/wiki/Totally_disconnected_space. This is equivalent to saying that every two points can be separated by a clopen, which is what you wrote. |
Apr 18 |
comment |
Minimal totally separated spaces
Why are you allowed to make the assumption that the topology of $X$ is generated by the clopen sets? |
Apr 18 |
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Minimal totally separated spaces
Your definition of "zero-dimensional" is actually known as totally disconnected. Zero-dimensionality is usually defined as "has a basis consisting of clopen sets". |
Apr 13 |
awarded | Nice Answer |
Apr 11 |
comment |
Axiom of choice for sets of finite sets
In the formulation of $AC(Z)$ you can drop the condition $x \neq \emptyset$ as it isn't doing anything. For the empty family you always have a choice function. |
Apr 1 |
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Mathematicians wearing hats on arbitrary total orders
Beautiful, and with a nicely hidden use of choice in the second sentence of the proof ;-) |
Mar 24 |
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About the proof of the proposition “there exists irrational numbers a, b such that a^b is rational”
It is sad that this is a research-level question. |
Mar 24 |
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About the proof of the proposition “there exists irrational numbers a, b such that a^b is rational”
I just feel like saying $(\sqrt{2})^{\log_2 9}$. But more seriously, it is unpedagocial to speak about what we know and do not know when explaining the form of a given proof. The proof in question is not constructive because it employs the inference rule of excluded middle -- and the proof remains non-constructive no matter what else you prove or know. However, the proof can be transformed to a different proof which is constructive (assuming there is a constructive proof of irrationality of $\sqrt{2}^{\sqrt{2}}$). |
Mar 12 |
awarded | Good Answer |
Mar 12 |
revised |
What does a theoretical mathematician do?
Removing gender pronouns because someone complained about them. |
Mar 9 |
comment |
A topological concept dual to compactness
Hmm, @PaulTaylor will come to the rescue here. Maybe I am mixing up external and internal definitions. In his setting we actually work internally with exponentials, so we require an "internal" right adjoint to $\Sigma^{!} : \Sigma^1 \to \Sigma^X$. This makes a difference because the said right adjoint needs to be continuous as well. Is that the problem? |
Mar 8 |
accepted | Why is the path fibration a strong Hurewicz fibration? |
Mar 7 |
comment |
Category of Gödel Codings? [Reference Request]
Categories of represented spaces are locally cartesian closed and regular. (I say "categories" because using the Baire space is just one option.) To get Borel realizability going we'd need to know whether there is a reasonable category of measurable spaces and measurable maps which is at least weakly cartesian closed. I thought about this once but did not get far, and my measure-theory colleagues refuse to think about "weakly cartesian closed". |
Mar 7 |
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Category of Gödel Codings? [Reference Request]
When I write that book on computable mathematics I will present classical computability as an arcane way of thinking about computation. I'll even add a hint of perversion to it. |
Mar 6 |
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Category of Gödel Codings? [Reference Request]
Thanks for pointing out that realizability also incorporates the continuous version, namely TTE and equilogical spaces (and domain-theoretic representations as well). It would be interesting to see if the Borel stuff is of the realizability kind as well. |
Mar 6 |
awarded | Enlightened |