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Difference between revisions of "Discrete topology"

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''on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331701.png" />''
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''on a set $X$''
  
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The topology in which every set is open (and therefore every set is closed). The discrete topology is the largest element in the lattice of all topologies on the given set. The term  "discrete topology" is sometimes understood in a somewhat wider sense, viz. as a topology in which intersections of arbitrary numbers of open sets are open. In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331702.png" />-spaces, both definitions coincide. In this sense, the theory of discrete spaces is equivalent to the theory of partially ordered sets.
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The topology in which every set is open (and therefore every set is closed). The discrete topology is the largest element in the lattice of all topologies on the given set.  If $X$ is a space with the discrete topology, then every map from $X$ to any other topological space is continuous.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Diskrete Räume" ''Mat. Sb.'' , '''2''' (1937) pp. 501–520</TD></TR></table>
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* J.L. Kelley, "General topology", Graduate Texts in Mathematics '''27''' Springer (1975) {{ISBN|0-387-90125-6}} {{MR|0370454}} {{ZBL|0306.54002}}
  
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[[Category:General topology]]
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====Comments====
 
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The equivalence alluded to above is obtained as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331703.png" /> is a pre-ordered set (cf. [[Pre-order|Pre-order]]), then define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331704.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331705.png" />. With the topology generated by the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331707.png" /> becomes a [[Discrete space|discrete space]].
 
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331708.png" /> is a discrete space, put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d0331709.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d03317010.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d03317011.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d03317012.png" />, defines a pre-order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d03317013.png" />.
 
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These constructions are each others inverses. Moreover, discrete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033170/d03317014.png" />-spaces correspond to partial orders and  "real"  discrete spaces correspond to discrete orders.
 
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This simple idea and variations thereof have proven to be extremely fruitful, see, e.g., [[#References|[a1]]].
 
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====References====
 
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<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></table>
 

Latest revision as of 18:49, 14 November 2023

on a set $X$

The topology in which every set is open (and therefore every set is closed). The discrete topology is the largest element in the lattice of all topologies on the given set. If $X$ is a space with the discrete topology, then every map from $X$ to any other topological space is continuous.

References

  • J.L. Kelley, "General topology", Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 MR0370454 Zbl 0306.54002
How to Cite This Entry:
Discrete topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_topology&oldid=11525
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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