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14h

comment 
Categorytheoretic characterization of zerodimensional spaces
In particular, "clopen sets separate points" can be encoded categorically very simply by saying "every monomorphism $2\to X$ has a left inverse", where $2$ is the coproduct of two copies of the terminal object. But I'm not sure if this is the sort of thing you're looking for... 
15h

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Categorytheoretic characterization of zerodimensional spaces
What would you consider "similar"? The category CHaus is rigid (every selfequivalence is naturally isomorphic to the identity), so in some sense every object in CHaus can be characterized categorically. 
17h

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Cyclic faithfully flat modules
This is trivially impossible for commutative rings (if $R/I$ is faithfully flat, tensor it with $0\to I\to R\to R/I\to 0$ to find that $I=0$). 
1d

revised 
Lattice Flatness Measure
edited tags 
2d

comment 
An exercise in the Kaplansky's book
More conceptually: if $i$ is an idempotent, any idempotent in $R/(i)$ lifts to an idempotent in $R$ (this is obvious when you think of idempotents as decompositions of the ring as a product). Any idempotent in $R/I$ is already idempotent in some $R/I_0$ where $I_0\subseteq I$ is generated by finitely many idempotents. Since the ideal generated by finitely many idempotents is just the ideal generated by their join, it follows that any idempotent in $R/I$ lifts to an idempotent in $R$. 
Aug
28 
revised 
Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
edited tags 
Aug
28 
awarded  Tag Editor 
Aug
28 
wiki  created latticetheory description 
Aug
28 
wiki  created latticetheory excerpt 
Aug
27 
comment 
A Linear Order from AP Calculus
@DmitryV: That's not what I was talking about (I was also thinking like a category theorist and not caring about that). If $f/g$ oscillates between approaching $0$ and approaching $\infty$, then $f\not\leq g$ and $g\not\leq f$. Actually, if $f$ and $g$ don't have to be positive, you could just have an unbounded set where $f$ is zero and $g$ is not and an unbounded set where $g$ is zero and $f$ is not. 
Aug
27 
comment 
A Linear Order from AP Calculus
This is not a total order; $f$ and $g$ can oscillate between $f\gg g$ and $g\gg f$. 
Aug
25 
comment 
Which algebraic relations are possible between algebraic conjugates?
Clearly the orbit of $\alpha$ when you iterate $f$ must be finite, which gives a simpler argument for your last two examples. 
Aug
25 
comment 
On the Riesz representation theorem II
This is Theorem 3.10 in Rudin's Functional Analysis. 
Aug
25 
answered  On the Riesz representation theorem II 
Aug
24 
answered  On the Riesz representation theorem 
Aug
24 
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On the Riesz representation theorem
Do you want that limit to just hold pointwise, or do you want something stronger? If you just want it pointwise, the Riesz representation theorem has nothing to do with thisyou can find such a net for any $\Phi$ (because you can find a $\phi$ that works for any finite set of $\psi$s). On the other hand, if you restrict to sequences, $\Phi$ does have to be bounded if $V$ is complete, but this is not obvious (it follows from the BanachSteinhaus theorem). 
Aug
20 
answered  Hedetniemi's conjecture for graphs with countable chromatic number 
Aug
19 
revised 
Can the KanThurston theorem be turned into some kind of equivalence between groups and spaces?
made title less generic 
Aug
19 
reviewed  Leave Open Can the KanThurston theorem be turned into some kind of equivalence between groups and spaces? 
Aug
18 
comment 
Reference for (co)limitpreserving functor $X\mapsto R^X$
@OwenBiesel: I don't know any reference for this exact argument (you can cite this very MO post if you want such an exact reference...), but many of the ideas involved are discussed in Peter Johnstone's book Stone spaces (in particular, some constructions with my functor $G$ for $R=\mathbb{Z}$ (but not its left adjoint) are discussed in section V.2 and my last paragraph is discussed in section VI.2.3) 