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Apr
17
comment Homology of infinite intersection
It seems like pretty much any interesting construction where the intersection isn't a finite CW-complex is likely to be a counterexample. For instance, the standard construction of the Cantor set by deleting intervals is a counterexample (more simply, an analogous construction of $\{0,1,1/2,1/3,1/4,\dots\}$ by deleting intervals from $[0,1]$ also works). One more promising definition of "nice" would be that both the $X_n$ and the intersection are all finite CW-complexes; I haven't been able to come up with a counterexample to that.
Apr
16
comment First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
Also, you really want to assume that the topology is semimetrizable, not just that it is first-countable (I don't know whether these are equivalent, but I doubt they are). Chris's construction is quite flexible and can easily be modified to be "resilient". For instance, you could take $X\coprod\mathbb{N}$, where adding elements of $X$ to anything but $0$ is the same as adding $1$.
Apr
16
comment First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
A monoid is resilient iff it has no idempotents besides the identity, since any idempotent $i$ is absorbing in the submonoid $\{1,i\}$ and any (locally) absorbing element is idempotent.
Apr
16
comment If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Chris is not saying that any topology on $M$ makes it a topological monoid, but that any topology on $X$ makes $M=X\coprod \{0,\infty\}$ a topological monoid. If I'm not mistaken, though, any semimetric on $X$ can be extended to a subinvariant semimetric on $M$ (just make $0$ and $\infty$ infinitely far away from $X$ and each other), so this will only give a counterexample in the case that $X$ is not semimetrizable at all.
Apr
16
comment If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
No need to apologize for answering your own question; it is the correct thing to do if you learn an answer later!
Apr
15
reviewed Reject suggested edit on nontrivial theorems with trivial proofs
Apr
15
reviewed Reject suggested edit on nontrivial theorems with trivial proofs
Apr
15
comment Simple groups and words
More generally, take anything in the normal subgroup of $F$ generated by $n$th powers.
Apr
13
comment If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
To elaborate on my last comment, every poset with the Alexandrov topology is canonically semimetrizable (let $d(x,y)=0$ if $x\leq y$ and $d(x,y)=1$ otherwise). Incidentally, every semilattice is also a topological monoid and this semimetric is subinvariant.
Apr
13
comment If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Do you know of a characterization of what spaces admit any semimetric structure at all? If I'm not mistaken, these are a lot more general than metrizable spaces, since the possibility of having $d(x,y)=0$ but $d(y,x)>0$ lets you get interesting non-Hausdorff topologies.
Apr
11
comment When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube
There are certainly quotients of compact groups that have the fixed point property (for instance, even-dimensional projective spaces).
Apr
7
comment Is the space of *-homomorphisms between two $C^*$-algebras locally path connected
You don't actually need $A$ to be commutative, because when $B$ is commutative all homomorphisms from $A$ to $B$ will factor through the Gelfand transform of $A$.
Apr
7
comment Is the space of *-homomorphisms between two $C^*$-algebras locally path connected
I believe your argument works when $B$ is commutative, but I don't see why it works when only $A$ is commutative. If the image of $A\to B$ is not central, then you may be able to conjugate it by a continuous family of unitaries to get a nontrivial path in $\mathrm{Hom}(A,B)$.
Apr
2
comment Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?
Very nice! I am reminded of a similar cardinality constraint I found in this answer.
Apr
2
answered Compactly supported cohomology of homotopy equivalent manifolds
Mar
30
comment Given a 2-category, is the hammock localization wrt the equivalences equivalent to taking the hom-wise nerve of the maximal subgroupoids?
If $C$ has one object and one 1-morphism, the first construction forgets about all the 2-morphisms but the second does not.
Mar
27
reviewed Reject suggested edit on Which Lie groups have adjoint representations that are bounded away from zero?
Mar
26
comment Automorphism group of compact abelian group
By Pontryagin duality, $X$ has the same automorphisms as its character group, which is discrete. In particular, for instance, the automorphisms of a torus are just $GL_n(\mathbb{Z})$.
Mar
25
comment Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?
The Kan loop group also has the advantage of actually being a simplicial group on the nose, rather than up to homotopy. On your related question, there is an unpointed version of the Kan loop group due to Dwyer and Kan.
Mar
25
comment Why doesn't choice imply global choice (in NBG)?
I think this is a fine question, but it's basically the same as this question that was migrated a few days ago.