A note on chains of open sets

Author:
John Ginsburg

Journal:
Proc. Amer. Math. Soc. **89** (1983), 317-325

MSC:
Primary 54A25; Secondary 03E35

DOI:
https://doi.org/10.1090/S0002-9939-1983-0712644-8

MathSciNet review:
712644

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained.

Theorem 1. *For every cardinal* *there exists a space* *in which all discrete subsets have cardinality at most* *and which contains a chain of* *open sets*.

Theorem 2. *If* *is regular and contains a chain of* *open sets, then* *contains a discrete subset of cardinality* .

Theorem 3. *Let* *denote the set of all maximal chains of open subsets of* *endowed with the Tychonoff topology*. (i) , *and* (ii) . *Here* *denotes the weight of the space* *and* *denotes the pseudocharacter of the space* .

**[1]**J. E. Baumgartner,*Chains and antichains in*, J. Symbolic Logic**45**(1980), 85-92. MR**560227 (81d:03054)****[2]**M. Bell and J. Ginsburg,*Chains and discrete sets in zero-dimensional compact spaces*, Proc. Amer. Math. Soc.**83**(1981), 149-152. MR**620002 (82h:54002)****[3]**-,*Spaces of chains*, in preparation.**[4]**P. Erdös and R. Rado,*A partition calculus in set theory*, Bull. Amer. Math. Soc.**62**(1956), 427-489. MR**0081864 (18:458a)****[5]**V. V. Fedorčuk,*The cardinality of hereditarily separable bicompacta*, Dokl. Akad. Nauk SSSR**222**(1975), 302-305. MR**0377797 (51:13966)****[6]**A. Hajnal and I. Juhasz,*Discrete subspaces of topological spaces*, Indag. Math.**29**(1967), 343-356. MR**0229195 (37:4769)****[7]**-,*Discrete subspaces of topological spaces*. II, Indag. Math.**31**(1969), 18-30. MR**0264585 (41:9177)****[8]**I. Juhasz,*Cardinal functions in topology*, Math. Centre Tracts 34, Math. Centre, Amsterdam, 1971. MR**0340021 (49:4778)****[9]**-,*Cardinal functions in topology--ten years later*, Math. Centre, Amsterdam, 1981.**[10]**W. Mitchell,*Aronszajn trees and the independence of the transfer property*, Ann. Math. Logic**5**(1972), 21-46. MR**0313057 (47:1612)****[11]**P. Zenor,*Hereditary**-separability and the hereditary**-Lindelof property in product spaces and function spaces*, Fund. Math.**106**(1980), 175-180. MR**584491 (82a:54039)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0712644-8

Keywords:
Chains of open sets,
discrete subset,
partially ordered set,
-generated,
maximal chains

Article copyright:
© Copyright 1983
American Mathematical Society