## 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
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### References:

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