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 -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 |
Discrete space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_space&oldid=54458