Questions tagged [compactifications]
The compactifications tag has no usage guidance.
122 questions
2
votes
1
answer
1k
views
Extending uniformly continuous functions on subspaces to non-metrizable compactifications
I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.
Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...
- 2,998
4
votes
2
answers
866
views
Non-uniqueness of smooth compactification
Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification?
In particular, suppose $X$ is a ...
- 1,040
2
votes
0
answers
574
views
tangent bundle of the toric variety of the wonderful compactification.
Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in $\...
- 3,472
11
votes
1
answer
434
views
A product on the square roots of unit matrix
There is a strange product that takes two square roots of unit matrix, say $A$ and $B$, $A^2=I$, $B^2=I$ to a square root again,
$$ A\star B=(A+B)^{-1}(A-B+2I), \qquad (A\star B)^2=I$$
Could anybody ...
- 605
0
votes
1
answer
811
views
Positive function with zero Haar integral
If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral degenerate?...
- 13
0
votes
2
answers
210
views
Locally compact, 0-dimensional, pseudocompact space
Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which ...
- 1,704
2
votes
2
answers
343
views
Does locally compact plus pseudocompact imply paracompact?
This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?
- 1,704
7
votes
2
answers
733
views
Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ remainder
We know that the space $\mathbb{R}$ has compactifications with one point remainder, and two point remainder. but there is no compactification of $\mathbb{R}$ with three point remainder and the same ...
- 1,788
7
votes
2
answers
2k
views
A question about some special compactifications of $\mathbb{R}$
We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a ...
- 1,788
3
votes
1
answer
459
views
When is the Freudenthal compactification an ANR?
Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What are ...
- 2,104
4
votes
0
answers
487
views
Embedding of a smooth variety into a complete smooth variety.
Consider the following fact from algebraic geometry:
Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set.
I know how to prove this fact ...
- 2,639
9
votes
2
answers
2k
views
Stone-Čech compactification of $\mathbb R$
Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
- 385
10
votes
2
answers
599
views
Non-bimeromorphic compactifications
Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...
- 7,902
5
votes
2
answers
1k
views
Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?
Quoting from http://en.wikipedia.org/wiki/End_(topology):
"Let X be a topological space, and suppose that
K1 ⊂ K2 ⊂ K3 ⊂ · · ·
is an ascending sequence of compact ...
- 53
14
votes
2
answers
4k
views
Direct construction of the Stone-Čech compactification using ultrafilters?
If $X$ is a set (regarded as a discrete space), its Stone-Čech compactification can be identified with the set of ultrafilters on $X$ with its natural (Stone) topology. If $X$ is a general ...
- 118k
8
votes
2
answers
2k
views
End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
- 3,033
7
votes
1
answer
578
views
Flatly compactifiable morphisms
Let $f:U \to S$ be a flat morphism. Let us say that $f$ is flatly compactifiable if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that
1) $U = X \...
- 39.3k
9
votes
1
answer
739
views
Compactifications of varieties with small complement
Let $X$ be a smooth variety over an algebraically closed field $k$. If it makes things easier, $X$ may be assumed to be quasi-projective. By Nagata (or quasi-projectivity) there exists a proper ...
- 4,450
12
votes
1
answer
1k
views
Reference Request for Drinfeld and Laumon Compactifications
Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
...
- 2,706
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
34
votes
6
answers
3k
views
Compactification theorem for differentiable manifolds ?
Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
- 23.3k
34
votes
2
answers
3k
views
"Transitivity" of the Stone-Cech compactification
Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...
- 114k