All Questions
8 questions
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
20
votes
2
answers
1k
views
The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,344
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
6
votes
2
answers
324
views
Nonvanishing section of infinite-dimensional tautological bundle
Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have ...
- 9,853
2
votes
0
answers
323
views
Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
1
vote
1
answer
134
views
Chain of interior of closed set
It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such ...
- 11
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
1
vote
0
answers
105
views
The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11