What is Compactification: Definition and 29 Discussions

In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".

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  1. S

    A Different symmetries or no symmetries in string theory?

    I was reading the book "A Fortunate Universe" by Geraint Lewis and Luke Barnes and something caught my attention: At page 195 the authors say that universes with different symmetries could be modeled and they would have dramatic results like having different conservation laws. I asked Mr...
  2. DuckAmuck

    A Compactification of spatial extra dimensions

    Hi everyone, I am looking at a paper on compact dimensions. Equation 65 makes sense except for the term of 4*pi*n*R in the denominator. Why is it 4*pi and not 2*pi? I cannot rationalize this. Please help. Thank you. https://arxiv.org/ftp/hep-ph/papers/0609/0609260.pdf
  3. nightingale123

    Topology: Determine whether a subset is a retract of R^2

    Homework Statement Let ##X=([1,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## and ##Y=((0,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## ##a)##Find subspaces of of the euclidean plane ##\mathbb{R}^2## which are homeomorphic to the compactification with one...
  4. A

    Interesting Effect of Conformal Compactification on Geodesic

    I'm trying to understand why timelike geodesics in Anti de-Sitter space are plotted as sinusoidal waves on a Penrose diagram (a nice example of the Penrose diagram for AdS is given in Figure 2.3 of this thesis: http://www.nbi.dk/~obers/MSc_PhD_files/MortenHolm_Christensen_MSc.pdf). Bearing in...
  5. S

    String Theory: Compactification Estimate Update

    A lot of people quote 10^ 500 as the number of different ways of compactifying the extra dimension in string theory. But I recall somoene saying at a talk that its now much larger than that. What's the current estimate?
  6. D

    Compactification of M Theory on Smooth G2 Manifolds

    I am currently reading the paper given here by Acharya+Gukov titled "M Theory and Singularities of Exceptional Holonomy Manifolds", and in particular right now am following section 4 where the field content of the effective 4-dimensional theory is derived by harmonic decomposition of the 11D...
  7. C

    Compactification effects in Kaluza-K theory - Gauge away EM?

    I was first considering to post this in the GR section on the forum, but as I've understood it, compactification is essential in string theory, so I thought that perhaps you guys know the subject better also in Kaluza-Klein theory. Compactification in Kaluza-Klein theory as I understand it is...
  8. R

    Understanding Minkowski Space Compactification

    Hi. I'm reading about the compactification of Minkowski Space, and there is a subject that is keeping me awake. They say that the group of conformal transformations is isomorphic to the group of pseudoorthogonal transformations with determinant equal to 1. I don't know how this happen and it...
  9. H

    Finding inverse for a homeomorphism on the sphere (compactification)

    hi there I'd like to show that the sphere \mathbb{S}^n := \{ x \in \mathbb{R}^{n+1} : |x|=1 \} is the one-point-compactification of \mathbb{R}^n (*) After a lot of trying I got this function: f: \mathbb{S}^n \setminus \{(0,...,0,1)\} \rightarrow \mathbb{R}^n (x_1,...,x_{n+1})...
  10. F

    What is the Definition of Compactification for a Topological Space?

    Hi guys, I am confused about the definition of compactification of a topological space. Suppose (X,τx) is a topological space. Define Y=X\cup{p} and a new topology τY such that U\subseteqY is open if (1) p \notin U and U\in \tauX or (2) p \in U and X-U is a compact closed subset of X...
  11. J

    One-point compactification of space of matrices with positive trace

    one-point compactification of space of matrices with non-negative trace Hi I'm a physicist and my question is a bit text-bookey but it is also part of the proof that the universe had a beginning...so could I ask anyway...You got q which is a continuous function of a 3 by 3 matrix where if any...
  12. M

    Minimal compactification of an infinite dimensional space

    Wikipedia seems fairly consistent in stating that infinite-dimensional topological vector spaces such as Hilbert space aren't locally compact, which means that they can't have a one-point compactification. As metric spaces they're Tychonoff spaces, and thus can be compactified with the...
  13. radou

    A compactification theorem issue

    So, in Munkres' chapter about the Stone-Čech compactification, there is a Lemma at the beginning of the chapter the proof of which seems a bit unclear to me. The Lemma states: Let X be a space and let h : X --> Z be an imbedding of X in the compact Hausdorff space Z. Then there exists a...
  14. radou

    One-point compactification problem

    Homework Statement Show that the one-point compactification of N (the naturals) is homeomorphic with the subspace {0} U {1/n : n is in N} of R. The Attempt at a Solution If we show that N is homeomorphic with {1/n : n is in N}, then this homeomorphism extends to the one-point...
  15. B

    N-dimensional torus compactification and string winding

    Hello, Im trying to learn about string theory in toroidal compactification on an undergraduate level. I am mostly using Zwiebach's "A first course in string Theory" but now I am trying to do something that doesn't seem to be covered in the book or any other literature explicitly. Perhaps...
  16. S

    What is the Stone-Cech compactification of a product of Boolean algebras?

    This is not an exercise from a textbook, but a question regarding a remark in a textbook, so I was not sure if this question belongs here or in the homework section. Forgive me if I have erred. I was reading Kusraev and Kutateladze, Boolean Valued Analysis. In it, the authors make the...
  17. P

    Geometric interpretation of a given Alexandrov compactification

    What is the Alexandrov compactification of the following set and give the geometric interpretation of it: [(x,y): x^2-y^2>=1, x>0] that is, the right part of the hyperbola along with the point in it. This is a question from my todays exam in topology. I wrote that the given set is...
  18. P

    STRACTWhy Do We Always Rescale the Null Direction in Conformal Compactification?

    Let's consider for example compactification of Minkowski spacetime or Kruskal extension of Schwartzschild. They are quite similar because in both cases we rescale the null direction. I wonder why we always rescale the null direction, not simply x or t.
  19. mnb96

    Distances, compactification & Möbius transformations

    Hi, I have two points on a one-dimensional Euclidean submanifold, say the x-axis. I want to assume that this subspace is kind of "cyclic". This is often accomplished with the compactification R\cup \{ \infty \} The question is: How can I compute distances (up to some constant factor)...
  20. M

    One point compactification of the positive integers

    How do we show the one point compactification of the positive integers is homeomorphic to the set K={0} U {1/n : n is a positive integer}? Say Y is the one point compactification of the positive integers. I know Y must contain Z+ and Y\Z+ is a single point. Also Y is a compact Hausdorff...
  21. W

    Compactification and Extension of maps.

    Hi: a couple of questions on (Alexandroff) 1-pt. compactification: Thanks to everyone for the help, and for putting up with my ASCII posting until I learn Latex (in the summer, hopefully.) I wonder if anyone still does any pointset topology. I see many people's eyes glace when I...
  22. S

    Can Compact Dimensions be Explained in Terms of Rudimentary Field Theory?

    I'm an undergrad just getting into lagrangian fields and symmetry, having my first advanced particle physics lecture tomorrow. I've read around the subject of high energy physics to some degree, but I'm very limited in what I can understand, largely due to the mathematical particulars that I...
  23. Jim Kata

    Calabi Yau compactification and M theory

    Ok, I really know nothing about this subject. Yeah, I know the definition, a Kahler manifold with a vanishing first chern class and a su(n) holonomy. I do think it's a cool mathematical concept that you could compact non-compact dimensions and in doing such you can generate the coupling...
  24. E

    Lightlike compactification

    I am trying to do problem 5 at the following website: http://www.myoops.org/twocw/mit/NR/rdonlyres/Physics/8-251String-Theory-for-UndergraduatesSpring2003/F4BA42A3-4DD9-402F-BDA8-6D5CB14B0FAF/0/ps1.pdf I got for (b) x' ~ \gamma \left( \gamma (1 -\beta^2) x'^0 + 2 \pi ( 1- \beta) R...
  25. E

    String theorists, I have a question about moduli stabilization and compactification

    Is string theory's position that the extra 6 dimensions are completely frozen, non-dynamical, static in their specific yau-calibi configuration, for all eternity, each frozen in exactly the same way at every point in space, never changing, while the 4 large dimensions are dynamical, according to...
  26. R

    Quantum Phase Compactification via Spacetime Expansion

    Non-relativistic string theory was introduced by d'Alembert, in 1747, with the first appearance of the wave equation and the d'Alembertian operator, which eventually became the foundations of "relativistic" field theories; for example, the theories of electromagnetism, special relativity...
  27. arivero

    Does the Inverse Square Law Justify Compactification to 3+1 Dimensions?

    I mean, beyond the physical argument... has people (scientists) worried about to justify why the configuration space of Nature should compactify to 3+1? In a previous thread, we found (we=marcus+me+orion+rest of readers if any) that gravitational coupling constant simplifies out of some...
  28. marcus

    The Bohr compactification of the Reals

    the Bohr compactification of the real line RBohr is essential (it seems) to Loop Quantum Cosmology. Here are some links illustrating that: https://www.physicsforums.com/showthread.php?s=&postid=147432#post147432 I want to understand the Bohr compactification better. It is putting a...
  29. R

    Models for Compactification from Inflation?

    Here is a good paper that is quite neat makes a good case for evolving extensions to Inflation models. http://arxiv.org/PS_cache/hep-th/pdf/0307/0307179.pdf