Difference between revisions of "Discrete space"
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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. |
| − | + | 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. | |
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| − | + | 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]]]. | |
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| − | This simple idea and variations thereof have proven to be extremely fruitful, see, e.g., [[#References|[ | ||
====References==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[ | + | <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 |
Discrete space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_space&oldid=37242