Questions tagged [compactifications]
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122 questions
13
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Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 11.4k
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0
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76
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About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 193
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1
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A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 11.4k
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2
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Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
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11
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2
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314
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Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
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0
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60
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Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$
Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.
However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
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13
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1
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329
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Is there a metric compactification that doesn't create new paths?
Every separable metric space $A$ has a metrizable compactification, i.e. a compact metrizable space $X$ for which $A$ embeds topologically as a dense subspace of $X$. There are many approaches to ...
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Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere
I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
- 19.6k
1
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0
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When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
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1
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Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly.
The Bohr compactification of $\mathbb R$ has two equivalent definitions.
The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 148k
8
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1
answer
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Is there an explicit construction of the Bohr Compactification of the Integers?
Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including ...
- 53.3k
3
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107
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Fulton-MacPherson compactifications (and wonderful compactifications) as relative Proj
Let $X$ be a (smooth complex projective) variety.
The Fulton-MacPherson compactification $X[n]$ is obtained from $X^n$ by blowing up the diagonals in a certain order. Is it possible to write down a (...
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2
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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?
- 4,713
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0
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Homeomorphism between interiors of simplex and permutohedron
The $n$-dimensional permutohedron $P_n$ is a polytope whose facets (i.e.\ codimension $1$ faces) are in 1-to-1 correspondence with all faces (of codimension${}\geq 1$) of the $n$-simplex $\Delta_n$, ...
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1
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Points in the Stone Cech compactification are intersection of open sets
Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
- 11.4k
4
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2
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Compactification of a product of manifolds
Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
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1
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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 ...
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What is the meaning of universal family of Fulton Macpherson configuration space?
Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process ...
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1
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204
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Are degrees and ramification degrees preserved upon passing to the smooth compactification?
Let $\phi ：C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification.
Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}...
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1
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0
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263
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Implicit function theorem and compactification of algebraic curve
Let $C$ be a singular curve defined over a local field $K$.
Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization).
Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
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34
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6
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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 [...
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0
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Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?
It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification).
My current state of belief/knowledge:
The ...
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0
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152
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Line bundles on toric varieties associated to Weyl chamber
I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
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1
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On the zero-dimensional strata of the Fulton-MacPherson conpactification
Let $\operatorname{Conf}_n(\mathbb{R})$ be the configuration space of $n$ marked points on the real line. What is the difference between $\operatorname{Conf}_n(\mathbb{R})$ and the locus of zero-...
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1
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What is the Freudenthal compactification of a wildly punctured n-sphere?
Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$.
Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
- 1,926
4
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3
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998
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How is a MacNeille completion "universal" like a beta-compactification is "universal"?
The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...
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6
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2
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830
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Do all homogeneous spaces have homogeneous compactifications?
Let $X$ be a separable metric space which is homogeneous, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.
A compactification of $X$ is a ...
CommunityBot
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0
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218
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Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
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0
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190
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Ends of a metric space?
I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse ...
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0
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Algebraic characterisation of the end space of a proper geodesic space in terms of non-continuous functions
$\DeclareMathOperator\Bf{B_\mathrm{f}}\DeclareMathOperator\Bc{B_\mathrm{c}}\DeclareMathOperator\Cf{C_\mathrm{f}}\DeclareMathOperator\Cd{C_\mathrm{d}}\DeclareMathOperator\Cc{C_\mathrm{c}}$Based on a ...
- 63.9k
1
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0
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143
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End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
- 106
1
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1
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Representing split-complex numbers as intervals and related compactification
Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
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2
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1
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Inducing maps between Martin boundaries
This is a reworking of a question I asked on math.se.
Given two countable discrete metric spaces $X_{1}$ and $X_{2}$, each equipped with a (irreducible and transient)* random walk given by transition ...
CommunityBot
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1
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1
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Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions
I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to $\beta\...
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Explicit toroidal compactification of Hilbert modular varieties
Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
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0
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153
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Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
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Is the one-point compactification of $\mathbb{N}$ computably countable?
The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of ...
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6
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1
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How complicated can the path component of a compact metric space be?
Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...
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Smooth toric compactification of $\mathbb C^n$
By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...
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Functoriality for compactifications of locally symmetric spaces
Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...
- 63.9k
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Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?
Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and $...
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Characterization of pretty compact spaces
This is a cross post from MSE.
I believe that the following problem have already been considered by some sophisticated topologist.
Definition 1. A non-compact Hausdorff topological space $X$ is called ...
- 11.4k
1
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1
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Fixed points of one-point-compactification
Let $M$ be a locally compact (Hausdorff) space, and $g:M\to M$ an isomorphism (think of an action of a finite cyclic group).
By some generalities one can show that the "obvious" map $(M^g)^+\...
- 151
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0
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Completion/Compactification of a Kähler metric on $\mathbb C^2$
Consider $\mathbb{C}^{2}$ equipped with the Kähler form
$$
\omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right),
$$
where $\mu$ is a positive real ...
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2
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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 ...
- 3,768
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1
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333
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Possible cardinalities of the remainders of compactifications of $\Bbb R$
With the usual topology on $\Bbb R$, a compactification $\mathrm{id}_{\Bbb R}:\Bbb R\to v\Bbb R$ can have a remainder $v\Bbb R \setminus \Bbb R$ of cardinality $1,2, 2^{\aleph_0}=\mathfrak c,$ or $2^{\...
- 18.5k
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2
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2k
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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 ...
- 4,713
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1
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Does a flat compactification always exist?
Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question ...
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0
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162
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A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
CommunityBot
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Intuition for the McGerty-Nevins compactification of quiver varieties
In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations
of the preprojective ...
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