User:Richard Pinch/sandbox-9

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 ${}^c$ is the Kuratowksi closure operator, via $$ d(x,A) = \begin{cases} 0 & \ \text{if} x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ . $$

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

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 (1991) 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