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$$J$$-spaces. (English) Zbl 0942.54020
In this article the legendary mathematician Ernest Michael introduces a new class of topological spaces and makes a thorough study of these spaces. All spaces are Hausdorff. A space $$X$$ is called a $$J$$-space if, whenever $$\{A,B\}$$ is a closed cover of $$X$$ with $$A\cap B$$ compact, then $$A$$ or $$B$$ is compact. A space $$X$$ is a strong $$J$$-space if every compact $$K\subset X$$ is contained in a compact $$L\subset X$$ with $$X\smallsetminus L$$ connected. Every strong $$J$$-space is a $$J$$-space. It is surprising that $$\mathbb{R}$$ is not a $$J$$-space but for $$n>1$$, $$\mathbb{R}^n$$ is a strong $$J$$-space! However, $$\mathbb{R}^+$$ is a strong $$J$$-space!
Main results: (1) Every compact space is a strong $$J$$-space. (2) If $$X$$ and $$Y$$ are connected and non-compact, then $$X\times Y$$ is a strong $$J$$-space. (3) A space $$X$$ is a $$J$$-space if and only if every boundary-perfect map $$f:X\to Y$$ onto a non-compact space $$Y$$ is perfect. (4) A metrizable space $$X$$ is a $$J$$-space if and only if every closed map $$f:X\to Y$$ onto a non-compact, metrizable space $$Y$$ is perfect. The result is true if “metrizable” is replaced by “locally compact and paracompact”. (5) A locally compact, non-compact space $$X$$ is a $$J$$-space if and only if the Alexandroff one point compactification of $$X$$ is equivalent to its Freudenthal compactification. (6) A non-compact completely regular space $$X$$ is a $$J$$-space if and only if its complement in every compactification $$Y$$ of $$X$$ is relatively connected in $$Y$$. (7) $$J$$-spaces, but not strong $$J$$-spaces are preserved by perfect images. (8) $$J$$-spaces are preserved by images under closed maps with paracompact domain and first-countable or locally compact range. (9) $$\mathbb{R}$$ is not the image under a closed map of any paracompact topological linear space different from $$\mathbb{R}$$. (10) $$J$$-spaces and strong $$J$$-spaces are preserved by monotone perfect pre-images. (11) Product of two connected (strong) $$J$$-spaces is a (strong) $$J$$-space. (12) Every component of a strong $$J$$-space is a strong $$J$$-space. The result is not true for $$J$$-spaces. The paper is beautifully written with numerous observations and examples and can be followed by anyone with a basic knowledge of topology.

##### MSC:
 54D30 Compactness 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54D05 Connected and locally connected spaces (general aspects) 54G99 Peculiar topological spaces
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