Spaces in which compact subsets are closed and the lattice of T 1 -topologies on a set

Ofelia Teresa Alas; Richard Gordon Wilson

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 4, page 641-652
  • ISSN: 0010-2628

Abstract

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We obtain some new properties of the class of KC-spaces, that is, those topological spaces in which compact sets are closed. The results are used to generalize theorems of Anderson [1] and Steiner and Steiner [12] concerning complementation in the lattice of T 1 -topologies on a set X .

How to cite

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Alas, Ofelia Teresa, and Wilson, Richard Gordon. "Spaces in which compact subsets are closed and the lattice of $T_1$-topologies on a set." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 641-652. <http://eudml.org/doc/248958>.

@article{Alas2002,
abstract = {We obtain some new properties of the class of KC-spaces, that is, those topological spaces in which compact sets are closed. The results are used to generalize theorems of Anderson [1] and Steiner and Steiner [12] concerning complementation in the lattice of $T_1$-topologies on a set $X$.},
author = {Alas, Ofelia Teresa, Wilson, Richard Gordon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {KC-space; $T_1$-complementary topology; $T_1$-independent; sequential space; KC-space; minimal KC-topology; -complementary topology; -independent; sequential space},
language = {eng},
number = {4},
pages = {641-652},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Spaces in which compact subsets are closed and the lattice of $T_1$-topologies on a set},
url = {http://eudml.org/doc/248958},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Alas, Ofelia Teresa
AU - Wilson, Richard Gordon
TI - Spaces in which compact subsets are closed and the lattice of $T_1$-topologies on a set
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 641
EP - 652
AB - We obtain some new properties of the class of KC-spaces, that is, those topological spaces in which compact sets are closed. The results are used to generalize theorems of Anderson [1] and Steiner and Steiner [12] concerning complementation in the lattice of $T_1$-topologies on a set $X$.
LA - eng
KW - KC-space; $T_1$-complementary topology; $T_1$-independent; sequential space; KC-space; minimal KC-topology; -complementary topology; -independent; sequential space
UR - http://eudml.org/doc/248958
ER -

References

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  1. Anderson B.A., A class of topologies with T 1 -complements, Fund. Math. 69 (1970), 267-277. (1970) MR0281140
  2. Anderson B.A., Stewart D.G., T 1 -complements of T 1 -topologies, Proc. Amer. Math. Soc. 23 (1969), 77-81. (1969) MR0244927
  3. Engelking R., General Topology, Heldermann Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  4. Fleissner W.G., A T B -space which is not Katětov T B , Rocky Mountain J. Math. 10 3 (1980), 661-663. (1980) Zbl0448.54021MR0590229
  5. Franklin S.P., Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115. (1965) Zbl0132.17802MR0180954
  6. Pelant J., Tkačenko M.G., Tkachuk V.V., Wilson R.G., Pseudocompact Whyburn spaces need not be Fréchet, submitted. 
  7. Shakhmatov D., Tkačenko M.G., Wilson R.G., Transversal and T 1 -independent topologies, submitted. 
  8. Simon P., On accumulation points, Cahiers Topologie Géom. Différentielle Catégoriques 35 (1994), 321-327. (1994) Zbl0858.54008MR1307264
  9. Smythe N., Wilkins C.A., Minimal Hausdorff and maximal compact spaces, J. Austral. Math. Soc. 3 (1963), 167-177. (1963) Zbl0163.17201MR0154254
  10. Steen L.A., Seebach J.A., Counterexamples in Topology, Second Edition, Springer Verlag, New York, 1978. Zbl0386.54001MR0507446
  11. Steiner A.K., Complementation in the lattice of T 1 -topologies, Proc. Amer. Math. Soc. 17 (1966), 884-885. (1966) MR0193033
  12. Steiner E.F., Steiner A.K., Topologies with T 1 -complements, Fund. Math. 61 (1967), 23-28. (1967) MR0230277
  13. Tkačenko M.G., Tkachuk V.V., Wilson R.G., Yaschenko I.V., No submaximal topology on a countable set is T 1 -complementary, Proc. Amer. Math. Soc. 128 1 (1999), 287-297. (1999) MR1616605
  14. Wilansky A., Between T 1 and T 2 , Amer. Math. Monthly 74 (1967), 261-266. (1967) MR0208557

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