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#REDIRECT [[Cardinal characteristic]]
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{{TEX|done}}{{MSC|54A25}}
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One of the [[cardinal characteristic]]s of a [[topological space]] $X$.  The least upper bound of the cardinalities of discrete subspaces of $X$.
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For a Hausdorff space $X$, the spread is related to the [[Density (of a topological space)|density]]  $d(X)$ and [[cardinality]] $|X|$ by results of Hajnal and Juhász:
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$$
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d(X) \le 2^{s(X)} \, ;
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$$
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$$
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|X| \le 2^{2^{s(X)}} \ .
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$$
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====References====
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* Mary Ellen Rudin, ''Lectures on Set Theoretic Topology'', American Mathematical Society (1975) {{ISBN|0-8218-1673-X}} {{ZBL|0318.54001}}
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* András Hajnal; István Juhász "Some results in set-theoretic topology" ''Sov. Math., Dokl.'' '''8''' (1967) 141-143  {{ZBL|0153.52102}}
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* András Hajnal; István Juhász "Discrete subspaces of topological spaces" ''Nederl. Akad. Wet., Proc., Ser. A'' '''70''' (1967) 343-356  {{ZBL|0163.17204}}

Latest revision as of 18:14, 14 November 2023

2020 Mathematics Subject Classification: Primary: 54A25 [MSN][ZBL]

One of the cardinal characteristics of a topological space $X$. The least upper bound of the cardinalities of discrete subspaces of $X$.

For a Hausdorff space $X$, the spread is related to the density $d(X)$ and cardinality $|X|$ by results of Hajnal and Juhász: $$ d(X) \le 2^{s(X)} \, ; $$ $$ |X| \le 2^{2^{s(X)}} \ . $$


References

  • Mary Ellen Rudin, Lectures on Set Theoretic Topology, American Mathematical Society (1975) ISBN 0-8218-1673-X Zbl 0318.54001
  • András Hajnal; István Juhász "Some results in set-theoretic topology" Sov. Math., Dokl. 8 (1967) 141-143 Zbl 0153.52102
  • András Hajnal; István Juhász "Discrete subspaces of topological spaces" Nederl. Akad. Wet., Proc., Ser. A 70 (1967) 343-356 Zbl 0163.17204
How to Cite This Entry:
Spread. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spread&oldid=42655
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