27 votes

Is there a general theory of "compactification"?

There's a distinction that I find striking but don't know how to formalize usefully or how to evaluate its importance: In algebraic geometry, moduli spaces get compactified, and this involves adding a ...
Andreas Blass's user avatar
18 votes
Accepted

Is the one-point compactification of $\mathbb{N}$ computably countable?

The answer is no. Suppose that there are computable functions $q$ and $s$ as you describe. Let $k$ be a program that performs the following task. It starts enumerating $1$s at the start of the ...
Joel David Hamkins's user avatar
14 votes
Accepted

Do all homogeneous spaces have homogeneous compactifications?

Since you want a connected example: A surface of infinite genus has no homogeneous compactification. Indeed first observe a dense locally compact subset has to be open. So the surface has to be open, ...
YCor's user avatar
  • 60.1k
12 votes
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One point compactification of $(\mathbb{C}^{\ast})^n$

If $\widehat X$ is the 1-point compactification of $X$, then there is a homeomorphism (for, say, locally compact Hausdorff spaces) $$ \widehat{X \times Y} \cong \widehat X \wedge \widehat Y $$ with ...
Tyler Lawson's user avatar
  • 51.1k
12 votes

Do all homogeneous spaces have homogeneous compactifications?

The countable discrete space $\omega$ is a counterexample. Suppose $Y$ is a homogeneous compactification of $\omega$, with $X \subset Y$ being homeomorphic to $\omega$. As $Y$ is infinite, it ...
Nate Eldredge's user avatar
10 votes
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Is each compactification of $\mathbb N$ soft?

Let $A=\{0,2,4,\dots\}$ be the even numbers and let $B=\{1,3,5,\dots\}$ be the odd numbers. Topologize $A\cup \beta B$ so that $A$ is a sequence limiting to a unique point in $\beta B \setminus B $. ...
James Hanson's user avatar
  • 10.3k
10 votes
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Characterization of pretty compact spaces

A partial answer: other examples of pretty compact spaces are uncountable powers of $\{0,1\}$ and $[0,1]$, and in general products of uncountably many non-trivial compact Hausdorff spaces. See Problem ...
KP Hart's user avatar
  • 9,795
9 votes
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Uniqueness of smooth compactification upto a smooth morphism

This desire fails on a much more basic level. A smooth morphism is flat, and you can't even have a flat morphism between two smooth compactifications. A flat morphism has equidimensional fibers. If ...
Sándor Kovács's user avatar
8 votes
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Sigma algebras on the Stone–Čech compactification of a countable discrete group

No. First of all, take note that for compact zero-dimensional spaces $X$, the $\sigma$-algebra generated by all clopen sets is precisely the $\sigma$-algebra of Baire sets (recall that in a ...
Joseph Van Name's user avatar
8 votes
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Possible cardinalities of the remainders of compactifications of $\Bbb R$

Every connected compact Hausdorff space of weight $\aleph_1$ is the remainder $v \mathbb R \setminus \mathbb R$ of some compactification of $\mathbb R$. In particular, $[0,1]^{\aleph_1}$ is the ...
Will Brian's user avatar
  • 17.4k
7 votes

The Stone-Čech compactification of a inverse system

Let $X_n$ be $\{k\in\mathbb{N}:k\ge n\}$ and let $f_n:X_{n+1}\to X_n$ be the inclusion map. The inverse limit of the system $\{X_n,f_n,\mathbb{N}\}$ is empty; the limit of the system $\{\beta X_n,\...
KP Hart's user avatar
  • 9,795
7 votes
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Compactification of 6d (2, 0) SCFT on 4-manifolds

You could have a look at https://arxiv.org/abs/1806.02470 and references therein. EDIT (taking into account the comment): compactification of the $\mathcal{N}=(2,0)$ 6d superconformal field theory on ...
user25309's user avatar
  • 6,810
6 votes
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Wonderful compactification

I am just writing my comment above as an answer, just in the special case that the parabolic is a Borel subgroup $B$. In this case, the reductive part $M$ is a maximal torus $T$ in $G$. Denote by $W\...
6 votes
Accepted

Minimal Nagata-like compactification

The following argument shows that the very best thing you could hope for does not work. I'm not sure whether I need all of my assumptions. Claim: There does not exist a compactification ...
R. van Dobben de Bruyn's user avatar
6 votes

Compactification theorem for differentiable manifolds ?

I'm just trying to give a partial solution to the second question. Those examples had been mentioned in the previous posts. That is, finding manifolds which are open interiors of compact manifolds ...
Shijie Gu's user avatar
  • 1,936
6 votes

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

A metric space $X$ is called isometrically homogeneous if for any points $x,y\in X$ there exists a bijective isometry $f:X\to X$ such that $f(x)=y$. For isomemtrically homogeneous spaces this ...
Taras Banakh's user avatar
  • 40.8k
6 votes
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Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Here is a partial answer: the Continuum Hypothesis implies that all Parovichenko spaces are soft-Parovichenko; the proof is a bit long, so I put it in a PDF-file on my website. Also, I retract my ...
KP Hart's user avatar
  • 9,795
5 votes
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Naive compactification of $\mathbb{C}^*$-fibrations

No that is not true. There is a much simpler example than in the MO answer above. I might have written the following example in one of my earlier MO answers. Let the proper target of the morphism ...
5 votes

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

Theorem 1.4 in C. Frances' preprint "Rigidity at the boundary for conformal structures and other Cartan geometries" asserts that the geodesic compactification is unique (up to conformal diffeomorphism)...
Jeffrey Case's user avatar
  • 1,483
5 votes

Sheaf (Gieseker) compactification of moduli space of vector bundles

This is perhaps more of a comment, but it's too long. I'm coming from the algebraic side of things, but it seems to me that there are several issues here. Be warned that when I say curve I mean a (...
Jack Huizenga's user avatar
5 votes

Nowhere compact subsets of the plane

According to the definition here a topological space is nowhere compact if every compact subset has empty interior. Assuming that's that you mean by "nowhere compact", the following set $X$ seems to ...
bof's user avatar
  • 11.5k
5 votes

Compactification of a product of manifolds

Yes, the quotient $C_M = \overline{M} \times \overline{\mathbb{R}} / \left\{\{x\} \times \overline{\mathbb{R}} : x \in \partial\overline{M}\right\}$ seems to do the job, where I mean that the points ...
Igor Khavkine's user avatar
4 votes
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Bohr compactification as a topological compactification

As Francois Ziegler answered, for a locally compact group $G$, the Bohr compactification of $G$ is a compactification in the usual sense iff $G$ is compact. This is true with no further restrictions. ...
Uri Bader's user avatar
  • 11.4k
4 votes

Bohr compactification as a topological compactification

The answer is no: unless $G$ is compact, its subspace topology inside $bG$ (called its Bohr topology) is much weaker than its own. So $\iota:G\to bG$, while continuous with dense image, is not an ...
Francois Ziegler's user avatar
4 votes

Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

There is, in general, a on-to-one correspondence between closed subalgebras of $C^*(X)$ (the algebra of bounded continuous real-valued functions) and the compactifications of $X$. Closed in the sense ...
KP Hart's user avatar
  • 9,795
4 votes
Accepted

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

The obvious (to me) counterexamples are $\beta \mathbb{N}$ and $\beta \mathbb{R}$ ( the Čech-Stone compactifications) which are non-completely normal compactifications (classic fact, see ...
Henno Brandsma's user avatar
4 votes
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Does a flat compactification always exist?

This already fails if $S$ is regular of dimension $3$ and $\pi$ is quasi-finite. Indeed, let $X$ be a normal affine variety over $\mathbf C$ of dimension $3$ with an isolated non-Cohen–Macaulay ...
R. van Dobben de Bruyn's user avatar
4 votes
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How complicated can the path component of a compact metric space be?

Not every path connected separable metric space is homeomorphic to a path component of a compact metric space. The following cardinality arguments can be used: Fact 1. There is up to homeomorphism ...
Benjamin Vejnar's user avatar
4 votes
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Are degrees and ramification degrees preserved upon passing to the smooth compactification?

The first claim is true. The degree can be defined as the degree of the extension on function fields $k(C_1)/k(C_2)$, but $k(\tilde{C}_1) = k(C_1)$ and $k(\tilde{C}_2)= k(C_2)$ so $k(C_1)/k(C_2) = k(\...
Will Sawin's user avatar
  • 135k
4 votes
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Points in the Stone Cech compactification are intersection of open sets

Yes if the point is from $\mathbb{N}$ (it is isolated). No if the point is in $\beta\mathbb{N}\setminus\mathbb{N}$ because in that subspace every nonempty $G_\delta$-set has nonempty interior, see ...
KP Hart's user avatar
  • 9,795

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