\(J\)-spaces.

*(English)*Zbl 0942.54020In 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.

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.

Reviewer: S.A.Naimpally (Nepean)

##### 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 |