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defining topological spaces

  • Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.
    In topology: Topological space

    …sets in T are called open sets and T is called a topology on X. For example, the real number line becomes a topological space when its topology is specified as the collection of all possible unions of open intervals—such as (−5, 2), (1/2, π), (0, Square root of√2), …. (An analogous…

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  • In Hausdorff space

    …specified collection of subsets, called open sets, that satisfy three axioms: (1) the set itself and the empty set are open sets, (2) the intersection of a finite number of open sets is open, and (3) the union of any collection of open sets is an open set. A Hausdorff…

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