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 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