Namespaces
Variants
Actions

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

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article: Core-compact space)
 
(move)
Line 5: Line 5:
  
 
A space is core compact if and only if the product of the identity with a quotient map is quotient.
 
A space is core compact if and only if the product of the identity with a quotient map is quotient.
 +
=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 field]]s, [[real closed field]]s 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.
 +
 +
====References====
 +
* Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 {{ZBL|1235.12002}}
 +
 +
=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 number]]s:
 +
$$
 +
\pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ .
 +
$$
 +
 +
The group has an action on the three-sphere with [[tetrahedral space]] as quotient.
 +
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 76  {{ZBL|0732.51002}}</TD></TR>
 +
</table>
 +
 +
=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====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 77  {{ZBL|0732.51002}}</TD></TR>
 +
</table>
 +
 +
=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}}

Revision as of 19:30, 3 December 2017

Core-compact space

Let $X$ be a topological space with $\mathfrak{O}_X$ the collection of open sets. If $U, V$ are open, we say that $U$ is compact in $V$ if every open cover of $V$ has a finite subset that covers $U$. 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 compact in $V$.

A space is core compact if and only if $\mathfrak{O}_X$ 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.

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.

References

  • Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 Zbl 1235.12002

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 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 Zbl 0732.51002

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
How to Cite This Entry:
Richard Pinch/sandbox-9. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-9&oldid=42405