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Apr
17
comment The significance of filtered colimits in homotopy theory
@MikeMiller The simplest example is $\prod H_i(X)/\oplus H_i(X)$, where $H_i$ is, say, $i$th singular homology with integral coefficients. It's not terrible to verify this is a homology theory, and it obviously vanishes on finite complexes but not, say, an infinite wedge of spheres of unbounded dimension.
Apr
17
answered fiber bundle in topological category and smooth category.
Apr
17
comment Quotient topology by identifying the boundary of a circle as one point
For your last comment, the left hand of figure 22.4 gives two example of saturated open sets, one of which is contained in $\{(x,y):x^2+y^2<1\}$ and one which contains $S^1$, precisely as you say.
Apr
17
answered The significance of filtered colimits in homotopy theory
Apr
17
comment Is this differential equation separable?
Yes, that's fine.
Apr
17
reviewed Looks OK Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid?
Apr
10
comment Vector bundle base space map
Well, you could take $B'$ a point and $E'$ the zero-dimensional bundle over it. Do your bundle maps have to preserve rank?
Apr
8
comment Quotient of a category by equality in Grothendieck group
$K_0$ of a commutative ring is the groupification of projective modules under direct sum. You might be thinking of the Picard group. Since all exact sequences of projective modules split, the relation you mention is automatic in the classical case, but must be imposed separately in a general abelian category.
Apr
8
comment Quotient of a category by equality in Grothendieck group
You might make explicit that you've abstracted from an abelian category with $\oplus$ to a general (symmetric) monoidal category-at least, I think that's what you've done.
Apr
7
comment Topological invariants by integrals
@Herrmann you sound really stressed out about all this! Maybe you should take a break.
Apr
7
comment If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?
I agree as well.
Apr
7
answered If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?
Apr
7
comment Distributor? Distributive analog of commutator and associator?
It's a fine idea, but as your side question hints I don't any examples in which it's useful.
Apr
7
comment Topological invariants by integrals
@Herrmann I look forward to seeing your more detailed answer!
Apr
6
comment If $X$ is quasi-projective but the scheme $\tilde{X}$ is affine, is $X$ necessarily affine?
@c_c_chaos the "schemification" of $X$, where we add points for all the closed irreducible subvarieties of $X$.
Apr
6
comment When is a number even?
You're right that $a^2$ is even because you can divide it by two. That's the very definition of "even."
Apr
5
answered Topological invariants by integrals
Apr
3
comment Axiomatization of Topology
To refine the comment about compactness, it's not the case that topologies can be characterized in terms of their compact sets. This is clear from the consideration of finite spaces, in which all subsets are compact. Similarly, connectedness can't distinguish a discrete space from a totally disconnected non-discrete space such as $\mathbf{Q}$. I'm still not sure what to say about the collection of dense subsets determining the topology.
Apr
3
comment Axiomatization of Topology
It's not true that you can start from 4. Not all topological spaces have enough real-valued functions. The spaces that most obviously can be so reconstructed are the compact Hausdorff spaces, via Gelfand duality. Neighborhoods, on the other hand, is a separate concept from openness.
Apr
3
comment Increasing sequence of open sets in a separable metric space.
@user1537366 thanks very much for the correction.