# zbMATH — the first resource for mathematics

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.
Reviewer: M.Ganster (Graz)

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54A35 Consistency and independence results in general topology
Full Text:
##### References:
  Arhangel’skii, A.V., A generic theorem in the theory of cardinal invariants of topological spaces, Comment. math. univ. carolin., 36, 305-325, (1995)  Arhangel’skii, A.V., Location type properties: relative strong pseudocompactness, Trudy mat. inst. RAN, 193, 28-30, (1992), (in Russian)  Arhangel’skii, A.V.; Genedi, H.M.M., Beginnings of the theory of relative topological properties, (), 3-48, (in Russian)  Arhangel’skii, A.V.; Genedi, H.M.M.; Arhangel’skii, A.V.; Genedi, H.M.M., Location of subspaces in spaces: relative versions of compactness, Lindelöf property, and separation axioms, Vestnik MGU ser. I, mat. mekh., Moscow univ. math. bull., 44, 6, 67-69, (1989), English translation: · Zbl 0745.54003  Arhangel’skii, A.V.; Gordienko, I.Ju., Locally finite topological spaces, Questions answers gen. topology, 12, 1, (1994)  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  Arhangel’skii, A.V.; Jaschenko, I.V., Relatively compact spaces and separation properties, Comm. math. univ. carolin., 37, 3, (1996) · Zbl 0851.54024  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  Chigogidze, A.Ch., On relative dimensions, (), 67-117, (in Russian) · Zbl 0625.54040  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  Engelking, R., General topology, () · Zbl 0684.54001  Gordienko, I.Ju., On relative topological properties of normality type, Vestnik moskov. un-ta ser. I, mat. mekh., 5, 77-78, (1992), (in Russian)  Gordienko, I.Ju., A characterization of relative Lindelöf property by relative paracompactness, (), 40-44, (in Russian) · Zbl 0878.54013  Grothendieck, A., Critéres de compacité dans LES espaces fonctionnels généraux, Amer. J. math., 74, 175-185, (1952) · Zbl 0046.11702  Michael, E., Another note on paracompact spaces, (), 822-828 · Zbl 0078.14805  Ranchin, D.V., On compactness modulo an ideal, Dokl. akad. nauk SSSR, 202, 761-764, (1972), (in Russian)  Stchepin, E.V., Real-valued functions, and spaces close to normal, Sibirsk. mat. zh., 13, 1182-1196, (1972), (in Russian)  Tkachuk, V.V., On relative small inductive dimension, Vestnik mosk. univ. ser. I, mat. mekh., 5, 22-25, (1982), (in Russian) · Zbl 0526.54020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.