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Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).
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Intuition behind Alexander duality
I like to think of Alexander duality in terms of linking numbers of submanifolds (or, in general, k cycles). This is one way to define the pairing you are looking for. In general, consider a $k$-cycle …