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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2019

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2019
    • (edited Oct 17th 2019)

    starting something, for the moment just so as to record this fact:

    Let XX be an non-empty regular topological space and n2n \geq 2 \in \mathbb{N}.

    Then the injection

    Conf n(X)exp n(X)/exp n1(X) Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)

    of the unordered configuration space of n points of XX into the quotient space of the space of finite subsets of cardinality n\leq n by its subspace of subsets of cardinality n1\leq n-1 is an open subspace-inclusion.

    Moreover, if XX is compact, then so is exp n(X)/exp n1(X)\exp^n(X)/\exp^{n-1}(X) and the inclusion exhibits the one-point compactification (Conf n(X)) +\big( Conf_n(X) \big)^{+} of the configuration space:

    (Conf n(X)) +exp n(X)/exp n1(X). \big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.

    v1, current

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