Namespaces
Variants
Actions

Difference between revisions of "Discrete space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(moving text from Discrete topology)
m (→‎References: isbn link)
 
(4 intermediate revisions by one other user not shown)
Line 1: Line 1:
 
In the narrow sense, a space with the [[discrete topology]].
 
In the narrow sense, a space with the [[discrete topology]].
  
In the broad sense, sometimes termed ''Alexandrov-discrete'', a [[topological space]] in which intersections of arbitrary families of open sets are open. In the case of $T_1$-spaces, both definitions coincide. In this sense, the theory of discrete spaces is equivalent to the theory of partially ordered sets.
+
In the broad sense, sometimes termed ''Alexandrov-discrete'', a [[topological space]] in which intersections of arbitrary families of open sets are open. In the case of $T_1$-spaces, both definitions coincide. In this sense, the theory of discrete spaces is equivalent to the theory of [[partially ordered set]]s.  If $(P,{\sqsubseteq})$ is a [[pre-order]]ed set, then define $O_x = \{ y \in P : y \sqsubseteq x \}$ for $x \in P$. With the topology generated by the sets $O_x$, $P$ becomes a discrete space.
  
====References====
+
If $X$ is a discrete space, put $O_x = \cap \{ O : x \in O, \  O \,\text{open} \}$ for $x \in X$. Then $y \sqsubseteq x$ if and only if $y \in O_x$, defines a pre-order on $X$, the [[specialization of a point]] pre-order.
<table>
 
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Diskrete Räume"  ''Mat. Sb.'' , '''2'''  (1937)  pp. 501–520 {{ZBL|0018.09105}}</TD></TR>
 
</table>
 
 
 
====Comments====
 
The equivalence alluded to above is obtained as follows: If $P$ is a pre-ordered set (cf. [[Pre-order]]), then define $O_x = \{ y \in P : y \le x \}$ for $x \in P$. With the topology generated by the sets $O_x$, $P$ becomes a [[Discrete space|discrete space]].
 
  
If $X$ is a discrete space, put $O_x = \cap \{ O : x \in O, \  O \text{open} \}$ for $x \in X$. Then $y \le x$ if and only if $y \in O_x$, defines a pre-order on $X$.
+
These constructions are mutually inverse. Moreover, discrete $T_0$-spaces correspond to partial orders and narrow-sense discrete spaces correspond to [[discrete order]]s.
  
These constructions are each others inverses. Moreover, discrete $T_0$-spaces correspond to partial orders and  "real"  discrete spaces correspond to discrete orders.
+
This simple idea and variations thereof have proven to be extremely fruitful, see, e.g., [[#References|[2]]].
 
 
This simple idea and variations thereof have proven to be extremely fruitful, see, e.g., [[#References|[a1]]].
 
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Gierz,  K.H. Hofmann,  K. Keimel,  J.D. Lawson,  M.V. Mislove,  D.S. Scott,  "A compendium of continuous lattices" , Springer  (1980)</TD></TR>
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Diskrete Räume"  ''Mat. Sb.'' , '''2'''  (1937)  pp. 501–520 {{ZBL|0018.09105}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  G. Gierz,  K.H. Hofmann,  K. Keimel,  J.D. Lawson,  M.V. Mislove,  D.S. Scott,  "A compendium of continuous lattices" , Springer  (1980) {{ISBN|3-540-10111-X}}  {{MR|0614752}}  {{ZBL|0452.06001}} </TD></TR>
 
</table>
 
</table>
  

Latest revision as of 18:49, 14 November 2023

In the narrow sense, a space with the discrete topology.

In the broad sense, sometimes termed Alexandrov-discrete, a topological space in which intersections of arbitrary families of open sets are open. In the case of $T_1$-spaces, both definitions coincide. In this sense, the theory of discrete spaces is equivalent to the theory of partially ordered sets. If $(P,{\sqsubseteq})$ is a pre-ordered set, then define $O_x = \{ y \in P : y \sqsubseteq x \}$ for $x \in P$. With the topology generated by the sets $O_x$, $P$ becomes a discrete space.

If $X$ is a discrete space, put $O_x = \cap \{ O : x \in O, \ O \,\text{open} \}$ for $x \in X$. Then $y \sqsubseteq x$ if and only if $y \in O_x$, defines a pre-order on $X$, the specialization of a point pre-order.

These constructions are mutually inverse. Moreover, discrete $T_0$-spaces correspond to partial orders and narrow-sense discrete spaces correspond to discrete orders.

This simple idea and variations thereof have proven to be extremely fruitful, see, e.g., [2].

References

[1] P.S. Aleksandrov, "Diskrete Räume" Mat. Sb. , 2 (1937) pp. 501–520 Zbl 0018.09105
[2] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
How to Cite This Entry:
Discrete space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_space&oldid=37242
This article was adapted from an original article by S.M. Sirota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article