Inner points and breadth in certain compact semilattices

Authors:
D. R. Brown and J. W. Stepp

Journal:
Proc. Amer. Math. Soc. **84** (1982), 581-587

MSC:
Primary 22A26; Secondary 22A15, 54H12

DOI:
https://doi.org/10.1090/S0002-9939-1982-0643754-0

MathSciNet review:
643754

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Abstract | References | Similar Articles | Additional Information

Abstract: A point is inner if there exists an open set containing such that for each open set with , the inclusion homomorphism : is nontrivial. In this note it is proved that, if is a compact, chainwise connected topological semilattice of codimension , and is a point of breadth , then is an inner point.

**[1]**L. W. Anderson,*On the breadth and**-dimension of a topological lattice*, Pacific J. Math.**9**(1959), 327-333. MR**0105465 (21:4206)****[2]**K. Baker and A. R. Stralka,*Compact, distributive lattices of finite breadth*, Pacific J. Math.**34**(1970), 311-320. MR**0282895 (44:129)****[3]**D. R. Brown,*Topological semilattices on the two-cell*, Pacific J. Math.**15**(1965), 36-46. MR**0176453 (31:725)****[4]**H. Cohen,*A cohomological definition of dimension for locally compact Hausdorff spaces*, Duke Math. J.**21**(1954), 209-224. MR**0066637 (16:609b)****[5]**G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. Scott,*A compendium of continuous lattices*, Heidelberg, 1980. MR**614752 (82h:06005)****[6]**K. H. Hofmann and P. S. Mostert,*Elements of compact semigroups*, Merrill, Columbus, Ohio, 1966. MR**0209387 (35:285)****[7]**R. J. Koch,*Arcs in partially ordered spaces*, Pacific J. Math.**9**(1959), 723-728. MR**0108553 (21:7269)****[8]**J. D. Lawson,*The relation of breadth and codimension in topological semilattices*, Duke Math. J.**37**(1970), 207-212. MR**0258687 (41:3333)****[9]**-,*The relation of breadth and codimension in topological semilattices*. II, Duke Math. J.**38**(1971), 555-559. MR**0282891 (44:125)****[10]**-,*A generalized version of the Vietoris-Begle theorem*, Fund. Math.**55**(1969), 65-72. MR**0248805 (40:2055)****[11]**J. D. Lawson and B. Madison,*Peripheral and inner points*, Fund. Math.**69**(1970), 253-266. MR**0275418 (43:1175)****[12]**-,*Peripherality in semigroups*, Semigroup Forum**1**(1970), 128-142. MR**0265504 (42:413)****[13]**E. H. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)****[14]**J. W. Stepp,*Semilattices which are embeddable in a product of min intervals*, Proc. Amer. Math. Soc.**28**(1971), 81-86. MR**0276147 (43:1895)****[15]**A. D. Wallace,*Acyclicity of compact connected semigroups*, Fund. Math.**50**(1961), 99-105. MR**0132533 (24:A2373)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0643754-0

Keywords:
Inner points,
codimension

Article copyright:
© Copyright 1982
American Mathematical Society