Questions tagged [compactifications]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
2 answers
333 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 vote
0 answers
92 views

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 $\Delta_n$, ...
1 vote
1 answer
123 views

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 ...
4 votes
2 answers
532 views

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 ...
3 votes
1 answer
450 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 ...
1 vote
0 answers
80 views

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 ...
0 votes
1 answer
197 views

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}...
1 vote
0 answers
251 views

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 \...
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 [...
1 vote
0 answers
91 views

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 ...
2 votes
0 answers
139 views

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}...
2 votes
1 answer
128 views

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-...
4 votes
1 answer
219 views

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 ...
4 votes
3 answers
938 views

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 ...
6 votes
2 answers
818 views

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 ...
2 votes
0 answers
198 views

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 ...
4 votes
0 answers
150 views

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 ...
3 votes
0 answers
57 views

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 ...
1 vote
0 answers
135 views

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 ...
0 votes
1 answer
157 views

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-...
2 votes
1 answer
148 views

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 ...
1 vote
1 answer
407 views

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\...
4 votes
0 answers
134 views

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 ...
0 votes
0 answers
136 views

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 $G$ ...
20 votes
1 answer
1k views

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 ...
6 votes
1 answer
295 views

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 ...
3 votes
0 answers
175 views

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 $\mathbb C^n$. ...
3 votes
0 answers
237 views

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 ...
5 votes
3 answers
395 views

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 $...
8 votes
1 answer
258 views

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 ...
1 vote
1 answer
212 views

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)^+\...
6 votes
0 answers
152 views

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 ...
7 votes
2 answers
704 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 ...
10 votes
1 answer
319 views

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^{\...
6 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 ...
8 votes
1 answer
417 views

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 ...
0 votes
0 answers
156 views

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 ...
5 votes
0 answers
262 views

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 ...
1 vote
1 answer
205 views

The Stone-Čech compactification of a inverse system

Is the Stone-Čech compactification of the inverse limit of an inverse system $\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces equal to the limit of the inverse system $\left\{ \beta X_{i},\beta ...
4 votes
1 answer
261 views

Nowhere compact subsets of the plane

Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty. Can $X$ be densely embedded into the plane? In other words, is there a dense set $X'\subseteq ...
36 votes
1 answer
3k views

Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
10 votes
2 answers
430 views

Which points in the Samuel compactification of a metric space $X$ are limits of uniformly discrete subsets of $X$?

Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow ...
1 vote
0 answers
161 views

Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
6 votes
1 answer
334 views

Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...
0 votes
1 answer
88 views

Does surjective map induce surjective map on Hewitt real compactifications?

Let $\beta X$ be the Stone-Čech compactification and $\upsilon X$ be the Hewitt real compactification of a completely regular space $X$. It is well known that any continuous surjective map $f:X\...
3 votes
1 answer
186 views

Research in compactifications of locally compact spaces

I would like to know how is it going the research in compactifications of locally compact Hausdorff spaces. Are there people doing this? Are there relevant conjectures on it?
3 votes
0 answers
190 views

Universal closure of schemes à la Nagata

Nagata compactification theorem is the following fundamental result: Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...
6 votes
1 answer
171 views

Is the conformal compactification of $M \setminus \{ p \}$ unique?

Let $(M,c)$ be a compact conformal manifold and $p \in M$. $M$ is a conformal compactification of $M \setminus \{ p \}$, because the embedding $M \setminus \{p\} \hookrightarrow M$ is an isometry. ...
2 votes
1 answer
72 views

Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?

The title is the question. Given a locally compact completely $T_{4}$ space $X$ (every subspace is $T_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely ...
2 votes
1 answer
122 views

Naive compactification of $\mathbb{C}^*$-fibrations

Let $\pi:X \to Y$ be a $\mathbb{C}^*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^*)^a$. Suppose ...