Explore Compactification through 10+ example sentences from English, with an explanation of the meaning. Ideal for language learners, writers and word enthusiasts.
Compactification meaning
- Any of various procedures of enlarging a topological space to make it compact.
- The space resulting from any such procedure.
- Any modification of a theory such that an infinite parameter becomes finite.
Using Compactification
- The main meaning on this page is: Any of various procedures of enlarging a topological space to make it compact. | The space resulting from any such procedure. | Any modification of a theory such that an infinite parameter becomes finite.
- In the example corpus, compactification often appears in combinations such as: one-point compactification, compactification of, of compactification.
Context around Compactification
- Average sentence length in these examples: 25.8 words
- Position in the sentence: 7 start, 8 middle, 5 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Compactification
- In this selection, "compactification" usually appears in the middle of the sentence. The average example has 25.8 words, and this corpus slice is mostly made up of statements.
- Around the word, alexandroff, serre and 186 stand out and add context to how "compactification" is used.
- Recognizable usage signals include one point compactification of the and one point compactification or alexandroff. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "compactification" sits close to words such as abdelaziz, absolutist and accrual, which helps place it inside the broader word index.
Example types with compactification
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. (15 words)
Here we describe gluing a pair of three-balls and then the one-point compactification. (15 words)
In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles. (17 words)
Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). (40 words)
Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8 and the G-index of the Dirac operator of the compact space must be nonzero. (40 words)
In this case it is called the one-point compactification or Alexandroff compactification of X. Recall from the above discussion that any compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification. (34 words)
Example sentences (20)
In this case it is called the one-point compactification or Alexandroff compactification of X. Recall from the above discussion that any compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.
For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification.
The one-point compactification In particular, the Alexandroff extension is a compactification of X if and only if X is Hausdorff, noncompact and locally compact.
There are a variety of compactifications, such as the Borel-Serre compactification, the reductive Borel-Serre compactification, and the Satake compactifications, that can be formed.
Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff).
Becker, Becker, and Schwarz 2007, p. 12 Number of dimensions main An example of compactification : At large distances, a two dimensional surface with one circular dimension looks one-dimensional.
But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point.
Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8 and the G-index of the Dirac operator of the compact space must be nonzero.
Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.
Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element to C 0 (X).
Greene 2000, p. 186 Compactification can be used to construct models in which spacetime is effectively four-dimensional.
Here we describe gluing a pair of three-balls and then the one-point compactification.
In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.
In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
In the late 1980s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.
Non-Hausdorff examples * The one-point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).
Note however that the projective plane RP 2 is not the one-point compactification of the plane R 2 since more than one point is added.
One-point compactification After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane.
Similarly, in Thermal quantum field theory a compactification of the euclidean time dimension leads to the Matsubara frequencies and thus to a discretized thermal energy spectrum.
Starting in 1991, a team of researchers including Michael Duff, Ramzi Khuri, Jianxin Lu, and Ruben Minasian considered a special compactification of string theory in which four of the ten dimensions curl up.
Common combinations with compactification
These word pairs occur most frequently in English texts:
- one-point compactification 12×
- compactification of 11×
- of compactification 4×
- alexandroff compactification 3×
- compactification or 2×
- borel-serre compactification 2×
- čech compactification 2×
- compactification at 2×
- compactification can 2×
- compactification is 2×