View source for Discrete space

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 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.  

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 $X$.

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

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

====References====
<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>
<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)</TD></TR>
</table>

{{TEX|done}}
[[Category:General topology]]
[[Category:Order, lattices, ordered algebraic structures]]