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**Relative topological properties and relative topological spaces.**
*(English)*
Zbl 0848.54016

This is a highly interesting survey article on so-called relative topological properties. It has to do with the question of how a given subspace \(Y\) of a space \(X\) is “located” in \(X\). The basic idea is as follows: with each topological property \(\mathbb{P}\) one can associate a relative version of it formulated in terms of location of \(Y\) in \(X\) in such a way that when \(Y\) coincides with \(X\) then this relative property coincides with \(\mathbb{P}\).

In the first part of this paper relative separation axioms are considered. Most results revolve around the situation when a subspace \(Y\) is normal (resp. strongly normal) in \(X\). \(Y \subseteq X\) is normal in \(X\) if for each pair \(A,B\) of closed disjoint subsets of \(X\) there are disjoint open subsets \(U\) and \(V\) in \(X\) such that \(A \cap Y \subseteq U\) and \(B \cap Y \subseteq V\). \(Y \subseteq X\) is strongly normal in \(Y\) if for each pair \(A,B\) of closed in \(Y\) disjoint subsets of \(Y\) there are open disjoint subsets \(U\) and \(V\) in \(X\) such that \(A \subseteq U\) and \(B \subseteq V\).

In the second part of the paper it is discussed how several relative compactness type properties influence relative separation properties. Typical results mentioned there are: (i) If \(Y \) is Lindelöf in a regular space \(X\) then \(Y\) is paracompact in \(X\), and (ii) if \(Y\) is paracompact in \(X\), and \(X\) is regular, then \(Y\) is normal in \(X\).

Throughout the paper a large number of still open problems are posed, thus setting the direction for possible future research.

In the first part of this paper relative separation axioms are considered. Most results revolve around the situation when a subspace \(Y\) is normal (resp. strongly normal) in \(X\). \(Y \subseteq X\) is normal in \(X\) if for each pair \(A,B\) of closed disjoint subsets of \(X\) there are disjoint open subsets \(U\) and \(V\) in \(X\) such that \(A \cap Y \subseteq U\) and \(B \cap Y \subseteq V\). \(Y \subseteq X\) is strongly normal in \(Y\) if for each pair \(A,B\) of closed in \(Y\) disjoint subsets of \(Y\) there are open disjoint subsets \(U\) and \(V\) in \(X\) such that \(A \subseteq U\) and \(B \subseteq V\).

In the second part of the paper it is discussed how several relative compactness type properties influence relative separation properties. Typical results mentioned there are: (i) If \(Y \) is Lindelöf in a regular space \(X\) then \(Y\) is paracompact in \(X\), and (ii) if \(Y\) is paracompact in \(X\), and \(X\) is regular, then \(Y\) is normal in \(X\).

Throughout the paper a large number of still open problems are posed, thus setting the direction for possible future research.

Reviewer: M.Ganster (Graz)

### MSC:

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

54A35 | Consistency and independence results in general topology |

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\textit{A. V. Arhangel'skii}, Topology Appl. 70, No. 2--3, 87--99 (1996; Zbl 0848.54016)

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### References:

[1] | Arhangel’skii, A. V., A generic theorem in the theory of cardinal invariants of topological spaces, Comment. Math. Univ. Carolin., 36, 305-325 (1995) |

[2] | Arhangel’skii, A. V., Location type properties: relative strong pseudocompactness, Trudy Mat. Inst. RAN, 193, 28-30 (1992), (in Russian) |

[3] | Arhangel’skii, A. V.; Genedi, H. M.M., Beginnings of the theory of relative topological properties, (General Topology. Spaces and Mappings (1989), MGU: MGU Moscow), 3-48, (in Russian) |

[4] | Arhangel’skii, A. V.; Genedi, H. M.M., Moscow Univ. Math. Bull., 44, 6, 67-69 (1989), English translation: · Zbl 0745.54003 |

[5] | Arhangel’skii, A. V.; Gordienko, I. Ju., Locally finite topological spaces, Questions Answers Gen. Topology, 12, 1 (1994) |

[6] | Arhangel’skii, A. V.; Gordienko, I. Ju., On relative normality and relative symmetrizability, Moscow Univ. Math. Bull., 50, 3, 28-31 (1995) · Zbl 0907.54012 |

[7] | Arhangel’skii, A. V.; Jaschenko, I. V., Relatively compact spaces and separation properties, Comm. Math. Univ. Carolin., 37, 3 (1996) · Zbl 0851.54024 |

[8] | Arhangel’skii, A. V.; Tartir, J., A characterization of compactness by a relative separation property, Questions Answers Gen. Topology, 14, 1 (1996) · Zbl 0851.54001 |

[9] | Chigogidze, A. Ch., On relative dimensions, (General Topology. Spaces of Functions and Dimension (1985), MGU: MGU Moscow), 67-117, (in Russian) · Zbl 0625.54040 |

[10] | Dow, A.; Vermeer, J., An example concerning the property of a space being Lindelöf in another, Topology Appl., 51, 255-260 (1993) · Zbl 0827.54014 |

[11] | Engelking, R., General Topology, (Sigma Series in Pure Mathematics, 6 (1989), Heldermann: Heldermann Berlin) · Zbl 0684.54001 |

[12] | Gordienko, I. Ju., On relative topological properties of normality type, Vestnik Moskov. Un-ta Ser. I, Mat. Mekh., 5, 77-78 (1992), (in Russian) |

[13] | Gordienko, I. Ju., A characterization of relative Lindelöf property by relative paracompactness, (General Topology. Spaces, Mappings and Functors (1992), MGU: MGU Moscow), 40-44, (in Russian) · Zbl 0878.54013 |

[14] | Grothendieck, A., Critéres de compacité dans les espaces fonctionnels généraux, Amer. J. Math., 74, 175-185 (1952) · Zbl 0046.11702 |

[15] | Michael, E., Another note on paracompact spaces, (Proc. Amer. Math. Soc., 8 (1957)), 822-828 · Zbl 0078.14805 |

[16] | Ranchin, D. V., On compactness modulo an ideal, Dokl. Akad. Nauk SSSR, 202, 761-764 (1972), (in Russian) |

[17] | Stchepin, E. V., Real-valued functions, and spaces close to normal, Sibirsk. Mat. Zh., 13, 1182-1196 (1972), (in Russian) |

[18] | Tkachuk, V. V., On relative small inductive dimension, Vestnik Mosk. Univ. Ser. I, Mat. Mekh., 5, 22-25 (1982), (in Russian) · Zbl 0526.54020 |

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