A Survey of J-spaces
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چکیده
This note is a survey of J-spaces. A space X is a J-space if, whenever {A, B} is a closed cover of X with A ∩ B compact, then A or B is compact. A space X is a strong J-space if every compact K ⊂ X is contained in a compact L ⊂ X with X\L connected. [As in [4], all maps are continuous and all spaces are Hausdorff.] 1.1. Every strong J-space X is a J-space. The two concepts coincide when X is locally connected, but in general (even for closed subsets of R 2) they do not. 2. Examples 2.1. A topological linear space X is a (strong) J-space if and only if X = R. 2.2. If X and Y are connected and non-compact, then X × Y is a strong J-space. 1 2.3. Let Y be a compact manifold with boundary B, and let A ⊂ B. Then Y \A is a (strong) J-space if and only if A is connected.
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