Let X be a topological space, and let A, B, and C be nonempty subsets of X such that A c B c C; that is, A is a proper subset of B, and B is a proper subset of C. Which of the following statements is (are) true? I. If A and C are connected, then B is connected. II. If A and C are compact, then B is compact. III. If A and C are Hausdorff, then B is Hausdorff.

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∻Solution by Steps∻ ‖ step 1 ⋮ Consider the given subsets $ A $, $ B $, and $ C $ of a topological space $ X $ with $ A \subset B \subset C $. ‖ ‖ step 2 ⋮ Analyze statement I: If $ A $ and $ C $ are connected, then $ B $ is connected. ‖ ‖ step 3 ⋮ Since $ A $ is connected and $ A \subset B $, any disconnection of $ B $ would lead to a disconnection of $ A $, which contradicts the assumption that $ A $ is connected. ‖ ‖ step 4 ⋮ Similarly, since $ B \subset C $ and $ C $ is connected, $ B $ cannot be disconnected without disconnecting $ C $. ‖ ‖ step 5 ⋮ Therefore, $ B $ must be connected if $ A $ and $ C $ are connected. ‖ ‖ step 6 ⋮ Analyze statement II: If $ A $ and $ C $ are compact, then $ B $ is compact. ‖ ‖ step 7 ⋮ Compactness is a topological property that is hereditary for closed subsets. ‖ ‖ step 8 ⋮ If $ B $ is a closed subset of $ C $, then $ B $ is compact because $ C $ is compact. However, without additional information about the closedness of $ B $, we cannot conclude that $ B $ is compact. ‖ ‖ step 9 ⋮ Analyze statement III: If $ A $ and $ C $ are Hausdorff, then $ B $ is Hausdorff. ‖ ‖ step 10 ⋮ The Hausdorff property is hereditary for all subsets, so if $ C $ is Hausdorff, then any subset of $ C $, including $ B $, is also Hausdorff. ‖ ∻Answer∻ ⚹ I and III are true. II cannot be determined to be true without additional information about the closedness of $ B $. ⚹ ∻Key Concept∻ ⚹Topological properties and hereditary nature in subsets⚹ ∻Explanation∻ ⚹Connectedness is preserved between nested subsets if the outer sets are connected. The Hausdorff property is hereditary for all subsets. Compactness is hereditary for closed subsets, but without knowing if $ B $ is closed in $ C $, we cannot conclude $ B $ is compact.⚹◊What is the definition of a proper subset in mathematics?⍭ Generate me a similar question◊

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