# Difference between revisions of "User:Richard Pinch/sandbox-9"

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+ | =Way below= | ||

+ | Let $(X,{\le})$ be a [[partially ordered set]]. The way below relationship $\ll$ determined by ${\le}$ is defined as $x \ll y$ if for each up directed subset $D$ of $X$ for which $y \le \sup D$, there is a $d \in D$ such that $x \le d$. Write $\Downarrow y = \{ x : x \ll y \}$. | ||

+ | |||

+ | A ''compact'' element $x \in X$ is one for which $x \ll x$. An ordered set is complete if $x = \sup\Downarrow x$ for all $x$. | ||

+ | |||

+ | ====References==== | ||

+ | * G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X {{MR|0614752}} {{ZBL|0452.06001}} | ||

+ | * Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications '''153''' Cambridge (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}} | ||

+ | |||

+ | =Relatively compact subset= | ||

+ | A subset $A$ of a [[topological space]] $X$ with the property that the [[Closure of a set|closure]] $\bar A$ of $A$ in $X$ is [[Compact space|compact]]. | ||

+ | |||

+ | A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$. | ||

+ | |||

+ | A space is compact if it is relatively compact in itself. | ||

+ | |||

+ | An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$. This formulation is equivalent to requiring that the set $A$ be [[way below]] $X$ with respect to set inclusion and the directed set of open subsets of $X$. | ||

+ | |||

+ | ====References==== | ||

+ | * N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 {{ZBL|1107.54002}} | ||

+ | * G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X {{MR|0614752}} {{ZBL|0452.06001}} | ||

+ | |||

=Core-compact space= | =Core-compact space= | ||

− | Let $X$ be a topological space | + | Let $X$ be a topological space. The space $X$ is core compact if for any $x \in X$ and open neighbourhood $N$ of $x$, there is an open set $V$ such that $N$ is [[Relatively-compact set|relatively compact]] in $V$ (every open cover of $V$ has a finite subset that covers $N$); equivalently, $N$ is [[way below]] $X$. |

+ | |||

+ | A space is core compact if and only if the collection of open sets $\mathfrak{O}_X$ is a [[continuous lattice]]. A [[locally compact space]] is core compact, and a [[sober space]] (and hence in particular a [[Hausdorff space]]) is core compact if and only if it is locally compact. | ||

− | A space is core compact if and only if | + | A space is core compact if and only if the product of the identity with a quotient map is quotient. The core compact spaces are precisely the exponentiable spaces in the [[category]] of topological spaces; that is, the spaces $X$ such that ${-} \times X$ has a right adjoint ${-}^X$. |

− | A | + | ====References==== |

+ | * Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications '''153''' Cambridge: Cambridge University Press (2014) (English) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}} | ||

=Developable space= | =Developable space= | ||

Line 39: | Line 64: | ||

d(x,A) = \inf\{ \delta(x,a) : a \in A \} \ . | d(x,A) = \inf\{ \delta(x,a) : a \in A \} \ . | ||

$$ | $$ | ||

− | and a topological space $X,{}^c$, where ${}^c$ | + | and a topological space $(X,{}^c)$, where ${}^c$ denotes the [[Kuratowksi closure operator]], via |

$$ | $$ | ||

− | d(x,A) = \begin{cases} 0 & \ \text{if} x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ . | + | d(x,A) = \begin{cases} 0 & \ \text{if}\ x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ . |

$$ | $$ | ||

+ | |||

+ | In the opposite direction, if $(X,d)$ is an approach space then the operation | ||

+ | $$ | ||

+ | A^c = \{ x \in X : d(x,A) < \infty \} | ||

+ | $$ | ||

+ | is a [[Čech closure operator]], giving $X$ the structure of a [[pre-topological space]]. However, the operation | ||

+ | $$ | ||

+ | A^C = \{ x \in X : d(x,A) = 0 \} | ||

+ | $$ | ||

+ | is a closure operator giving a topological structure on $X$. | ||

====References==== | ====References==== | ||

* Hofmann, Dirk (ed.); Seal, Gavin J. (ed.); Tholen, Walter (ed.) "Monoidal topology. A categorical approach to order, metric, and topology" Cambridge University Press (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}} | * Hofmann, Dirk (ed.); Seal, Gavin J. (ed.); Tholen, Walter (ed.) "Monoidal topology. A categorical approach to order, metric, and topology" Cambridge University Press (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}} | ||

− | + | * R. Lowen, "Approach spaces - a common supercategory of TOP and MET." ''Math. Nachr.'' '''141''' (1989) 183-226 {{ZBL|0676.54012}} | |

+ | * R. Lowen, "Index Analysis: Approach Theory at Work", Springer (2015) ISBN 1-4471-6485-7 {{ZBL|1311.54002}} | ||

=Ample field= | =Ample field= | ||

Line 52: | Line 88: | ||

A field $K$ is ample if and only if every absolutely irreducible curve over $K$ with a simple $K$-point has infinitely many $K$-points. | A field $K$ is ample if and only if every absolutely irreducible curve over $K$ with a simple $K$-point has infinitely many $K$-points. | ||

+ | |||

+ | If $K$ is ample, then the [[Galois theory, inverse problem of|inverse Galois problem]] for $K(T)$ is solved: every finite group occurs as a Galois group over $K(T)$. | ||

====References==== | ====References==== | ||

* Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 {{ZBL|1235.12002}} | * Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 {{ZBL|1235.12002}} | ||

+ | * Pierre Dèbes, Bruno Deschamps, "The regular inverse Galois problem over large fields" ''in ''Schneps, Leila (ed.) et al., Geometric Galois actions '''2'''", LMS Lecture Notes '''243''' Cambridge (1997) pp119-138 {{ZBL|0905.12004}} | ||

=Binary tetrahedral group= | =Binary tetrahedral group= | ||

Line 159: | Line 198: | ||

====References==== | ====References==== | ||

<table> | <table> | ||

− | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press ( | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X {{ZBL|0732.51002}}</TD></TR> |

</table> | </table> | ||

+ | |||

+ | =Scott topology= | ||

+ | A topology on a partially ordered set $(X,{\le})$ for which the open sets are the ''Scott open'' subsets: a downset $U$ is Scott open if for any set $S$ of $X$ with $\wedge S \in U$ then $\wedge F \in U$ for some finite $F \subseteq S$. | ||

+ | |||

+ | A function between partially ordered sets is Scott continuous in the Scott topologies if and only if it preserves meets of down directed sets. | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | * G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X {{MR|0614752}} {{ZBL|0452.06001}} | ||

+ | * Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications '''153''' Cambridge (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}} | ||

+ | |||

+ | =Compactly generated space= | ||

+ | ''Kelley space, $k$-space'' | ||

+ | |||

+ | A [[Hausdorff space|Hausdorff]] [[topological space]] in which a subset is closed if its intersection with any compact subset is closed. Every [[locally compact space|locally compact]] Hausdorff space is compactly generated, as is every [[First axiom of countability|first countable]] Hausdorff space. | ||

+ | |||

+ | The category of compactly generated spaces and continuous maps is equivalent to the category of Hausdorrf spaces and [[compactly continuous map]]s. | ||

+ | |||

+ | See: [[Exponential law (in topology)]]. | ||

+ | |||

+ | ====References==== | ||

+ | * Francis Borceux, "Handbook of Categorical Algebra: Volume 2, Categories and Structures", Encyclopedia of Mathematics and its Applications, Cambridge University Press (1994) ISBN 0-521-44179-X {{ZBL|1143.18002}} |

## Latest revision as of 20:01, 17 December 2017

## Contents

- 1 Way below
- 2 Relatively compact subset
- 3 Core-compact space
- 4 Developable space
- 5 Approach space
- 6 Ample field
- 7 Binary tetrahedral group
- 8 Binary icosahedral group
- 9 Binary octahedral group
- 10 Dodecahedral space
- 11 Étale algebra
- 12 Unit quaternion
- 13 Dicyclic group
- 14 Scott topology
- 15 Compactly generated space

# Way below

Let $(X,{\le})$ be a partially ordered set. The way below relationship $\ll$ determined by ${\le}$ is defined as $x \ll y$ if for each up directed subset $D$ of $X$ for which $y \le \sup D$, there is a $d \in D$ such that $x \le d$. Write $\Downarrow y = \{ x : x \ll y \}$.

A *compact* element $x \in X$ is one for which $x \ll x$. An ordered set is complete if $x = \sup\Downarrow x$ for all $x$.

#### References

- G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
- Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications
**153**Cambridge (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001

# Relatively compact subset

A subset $A$ of a topological space $X$ with the property that the closure $\bar A$ of $A$ in $X$ is compact.

A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$.

A space is compact if it is relatively compact in itself.

An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$. This formulation is equivalent to requiring that the set $A$ be way below $X$ with respect to set inclusion and the directed set of open subsets of $X$.

#### References

- N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 Zbl 1107.54002
- G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001

# Core-compact space

Let $X$ be a topological space. The space $X$ is core compact if for any $x \in X$ and open neighbourhood $N$ of $x$, there is an open set $V$ such that $N$ is relatively compact in $V$ (every open cover of $V$ has a finite subset that covers $N$); equivalently, $N$ is way below $X$.

A space is core compact if and only if the collection of open sets $\mathfrak{O}_X$ is a continuous lattice. A locally compact space is core compact, and a sober space (and hence in particular a Hausdorff space) is core compact if and only if it is locally compact.

A space is core compact if and only if the product of the identity with a quotient map is quotient. The core compact spaces are precisely the exponentiable spaces in the category of topological spaces; that is, the spaces $X$ such that ${-} \times X$ has a right adjoint ${-}^X$.

#### References

- Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications
**153**Cambridge: Cambridge University Press (2014) (English) ISBN 978-1-107-06394-5 Zbl 1297.18001

# Developable space

A **development** in a topological space $X$ is a sequence of open covers $G_n$ such that for all points $x \in X$ the stars
$$
\mathrm{St}(x,G_n) = \cup \{ U \in G_n : x \in U \}
$$
form a local base for $x$. A **developable space** is a space with a development. A metric space is a developable space: the sequence of collections of open balls of radius $1/n$ forming a development. A **Moore space** is a regular space with a development. A collection-wise normal Moore space is metrizable.

A **regular development** has the further property that if $U,V \in G_{n+1}$ with $U \cap V \neq \emptyset$, then there is $W \in G_n$ with $U \cup V \subset W$. Alexandroff and Urysohn proved that a space is metrizable if and only if it has a regular development.

#### References

- Alexandroff, P.; Urysohn, P. "Une condition nécessaire et suffisante pour qu’une classe $(\mathcal{L})$ doit une classe $(\mathcal{B})$",
*Comptes Rendus***177**(1923) 1274-1276. [1] Zbl 49.0702.06 Zbl 50.0696.01 - Bing, R.H. "Metrization of topological spaces",
*Canad. J. Math.***3**(1951) 175-186 DOI 10.4153/CJM-1951-022-3 Zbl 0042.41301

# Approach space

A generalisation of the concept of metric space, formalising the notion of the distance from a point to a set. An approach space is a set $X$ together with a function $d$ on $X \times \mathcal{P}X$, where $\mathcal{P}X$ is the power set of $X$, talking values in the extended positive reals $[0,\infty]$, and satisfying $$ d(x,\{x\}) = 0 \ ; $$ $$ d(x,\emptyset) = \infty \ ; $$ $$ d(x,A\cup B) = \min(d(x,A),d(x,B)) \ ; $$ $$ d(x,A) \le d(x,A^u) + u \ ; $$ where for $u \in [0,\infty]$, we write $A^u = \{x \in X : d(x,A) \le u \}$.

A metric space $(X,\delta)$ has an approach structure via $$ d(x,A) = \inf\{ \delta(x,a) : a \in A \} \ . $$ and a topological space $(X,{}^c)$, where ${}^c$ denotes the Kuratowksi closure operator, via $$ d(x,A) = \begin{cases} 0 & \ \text{if}\ x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ . $$

In the opposite direction, if $(X,d)$ is an approach space then the operation $$ A^c = \{ x \in X : d(x,A) < \infty \} $$ is a Čech closure operator, giving $X$ the structure of a pre-topological space. However, the operation $$ A^C = \{ x \in X : d(x,A) = 0 \} $$ is a closure operator giving a topological structure on $X$.

#### References

- Hofmann, Dirk (ed.); Seal, Gavin J. (ed.); Tholen, Walter (ed.) "Monoidal topology. A categorical approach to order, metric, and topology" Cambridge University Press (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001
- R. Lowen, "Approach spaces - a common supercategory of TOP and MET."
*Math. Nachr.***141**(1989) 183-226 Zbl 0676.54012 - R. Lowen, "Index Analysis: Approach Theory at Work", Springer (2015) ISBN 1-4471-6485-7 Zbl 1311.54002

# Ample field

A field which is existentially closed in its field of formal power series. Examples include pseudo algebraically closed fields, real closed fields and Henselian fields.

A field $K$ is ample if and only if every absolutely irreducible curve over $K$ with a simple $K$-point has infinitely many $K$-points.

If $K$ is ample, then the inverse Galois problem for $K(T)$ is solved: every finite group occurs as a Galois group over $K(T)$.

#### References

- Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 Zbl 1235.12002
- Pierre Dèbes, Bruno Deschamps, "The regular inverse Galois problem over large fields"
*in*Schneps, Leila (ed.) et al., Geometric Galois actions**2**", LMS Lecture Notes**243**Cambridge (1997) pp119-138 Zbl 0905.12004

# Binary tetrahedral group

The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as: $$ \langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ . $$ It is finite of order 24. It has the alternating group $A_4$ as quotient by the centre and the quaternion group of order 8 as a quotient.

This group may be realised as the group of invertible Hurwitz numbers: $$ \pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ . $$

The group has an action on the three-sphere with tetrahedral space as quotient.

#### References

[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 76 ISBN 0-521-20125-X Zbl 0732.51002 |

# Binary icosahedral group

The group $\langle 5,3,2 \rangle$ abstractly presented as: $$ \langle A,B \ |\ A^5=B^3=(AB)^2 \rangle \ . $$ It is finite of order 120.

The group has an action on the three-sphere with dodecahedral space as quotient.

#### References

[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 77 ISBN 0-521-20125-X Zbl 0732.51002 |

# Binary octahedral group

The group $\langle 4,3,2 \rangle$ abstractly presented as: $$ \langle A,B \ |\ A^4=B^3=(AB)^2 \rangle \ . $$ It is finite of order 48. It has the binary tetrahedral group $G_4 = \langle 3,3,2 \rangle$ as a subgroup of index 2.

The group has an action on the three-sphere with octahedral space as quotient.

#### References

[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 77 ISBN 0-521-20125-X Zbl 0732.51002 |

# Dodecahedral space

The result of identifying opposite faces of a dodecahedron by a right-handed turn of angle $\pi/5$. It is the quotient of the three-sphere by the binary icosahedral group.

Dodecahedral space is a homology sphere (Poincaré sphere).

#### References

- José Maria Montesinos, "Classical tessellations and three-manifolds" Springer (1987) ISBN 3-540-15291-1 Zbl 0626.57002

# Étale algebra

A commutative algebra $A$ finite-dimensional over a field $K$ for which the bilinear form induced by the trace $$ \langle x,y \rangle = \mathrm{tr}_{A/K} (x\cdot y) $$ is non-singular. Equivalently, an algebra which is isomorphic to a product of field $A \sim K_1 \times \cdots \times K_r$ with each $K_i$ an extension of $K$.

Since $\langle xy,z \rangle = \mathrm{tr}(xyz) = \langle x,yz \rangle$, an étale algebra is a Frobenius algebra over $K$.

#### References

- Tsit-Yuen Lam, "Lectures on Modules and Rings" Graduate Texts in Mathematics
**189**Springer (2012) ISBN 1461205255 Zbl 0911.16001

# Unit quaternion

A quaternion with norm 1, that is, $x_i + yj + zk + t$ with $x^2+y^2+z^2+t^2 = 1$.

The real unit quaternions form a group isomorphic to the special unitary group $\mathrm{SU}_2$ over the complex numbers, and to the spin group $\mathrm{Sp}_3$. They double cover the rotation group $\mathrm{SO}_3$ with kernel $\pm 1$.

The finite subgroups of the unit quaternions are given by group presentations $$ A^p = B^q = (AB)^2 $$ with $1/p + 1/q > 1/2$, denoted $\langle p,q,2 \rangle$. They are

- the cyclic groups $C_n$, , corresponding to $\langle n,n,1 \rangle$;
- the dicyclic groups, corresponding to $\langle n,2,2 \rangle$;
- the binary tetrahedral group $\langle 3,3,2 \rangle$;
- the binary octahedral group $\langle 4,3,2 \rangle$;
- the binary icosahedral group $\langle 5,3,2 \rangle$.

#### References

[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) Zbl 0732.51002 |

# Dicyclic group

A finite group of order $4n$, obtained as the extensions of the cyclic group of order $2$ by a cyclic group of order $2n$. It has the presentation $\langle n,2,2 \rangle$ and group presentation $$ A^n = B^2 = (AB)^2 \ . $$ It may be realised as a subgroup of the unit quaternions.

The dicyclic group $n=2$ is the quaternion group of order $8$.

#### References

[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002 |

# Scott topology

A topology on a partially ordered set $(X,{\le})$ for which the open sets are the *Scott open* subsets: a downset $U$ is Scott open if for any set $S$ of $X$ with $\wedge S \in U$ then $\wedge F \in U$ for some finite $F \subseteq S$.

A function between partially ordered sets is Scott continuous in the Scott topologies if and only if it preserves meets of down directed sets.

#### References

- G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
- Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications
**153**Cambridge (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001

# Compactly generated space

*Kelley space, $k$-space*

A Hausdorff topological space in which a subset is closed if its intersection with any compact subset is closed. Every locally compact Hausdorff space is compactly generated, as is every first countable Hausdorff space.

The category of compactly generated spaces and continuous maps is equivalent to the category of Hausdorrf spaces and compactly continuous maps.

See: Exponential law (in topology).

#### References

- Francis Borceux, "Handbook of Categorical Algebra: Volume 2, Categories and Structures", Encyclopedia of Mathematics and its Applications, Cambridge University Press (1994) ISBN 0-521-44179-X Zbl 1143.18002

**How to Cite This Entry:**

Richard Pinch/sandbox-9.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-9&oldid=42432