On maximal, minimal open and closed sets
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We obtain some conditions for disconnectedness of a topological space in terms of maximal and minimal open sets, and some similar results in terms of maximal and minimal closed sets along with interrelations between them. In particular, we show that if a space has a set which is both maximal and minimal open, then either this set is the only nontrivial open set in the space or the space is disconnected. We also obtain a result concerning a minimal open set on a subspace.
Abstract
We obtain some conditions for disconnectedness of a topological space in terms of maximal and minimal open sets, and some similar results in terms of maximal and minimal closed sets along with interrelations between them. In particular, we show that if a space has a set which is both maximal and minimal open, then either this set is the only nontrivial open set in the space or the space is disconnected. We also obtain a result concerning a minimal open set on a subspace.
참고문헌 (4)
- F. Nakaoka, "Some applications of minimal open sets", Int. J. Math. Math. Sci, 27(8), pp.471~476 (2001)
- F. Nakaoka, "Some properties of maximal open sets", Int. J. Math. Math. Sci, 2003(21), pp.1331~1340 (2003)
- F. Nakaoka, "Minimal closed sets and maximal closed sets", Int. J. Math. Math. Sci, pp.1~8 (2006)
- S. Willard, "General Topology" (1970), Addison-Wesley Publishing Company