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nLab > Latest Changes: short map

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  1. Is Met really Complete? Wikipedia argues that not, since indeed |R^|R isn’t even First-Countable so let alone Metrizable, when it would be the induced Topological Space underlying an Uncountable Product in Met…

    GrothenDitQue:

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 31st 2019

    This may depend on some details of what is meant by “metric space”; in particular, whoever wrote that might well have meant “Lawvere metric space”.

    A Lawvere metric space is a set XX equipped with a function d:X×X→[0,∞]d: X \times X \to [0, \infty] such that d(x,y)+d(y,z)≥d(x,z)d(x, y) + d(y, z) \geq d(x, z) and 0≥d(x,x)0 \geq d(x, x). Note that the value ∞\infty is allowed as a distance.

    If {X i} i∈I\{X_i\}_{i \in I} is a family of Lawvere metric spaces, then their cartesian product in the category of Lawvere metric spaces and short maps is given by the set ∏ i∈IX i\prod_{i \in I} X_i equipped with the distance

    d((x i),(y i))=sup id(x i,y i)d((x_i), (y_i)) = \sup_i d(x_i, y_i)

    In any event, you have to be a little careful in interpreting these things. Whatever notion of metric space is used, letting the morphisms be continuous maps between metric spaces is very different from letting the morphisms be something more closely tied to the metrics (e.g., short maps, Lipschitz maps), and the fact that classical metric/metrizable spaces are not closed under topological products is completely different from the fact that products may be perfectly sensible when we change the notion of morphism.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 1st 2019

    Clarified the discussion of products.

    diff, v11, current

  4. Added section on injective objects in Met ordMet_{ord}.

    diff, v13, current

  5. Added reference to lecture notes of Anton Petrunin.

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorVictor Sannier
    • CommentTimeDec 18th 2023

    Changed some notations, added another common name for short maps, and expanded the section on the category of metric spaces

    diff, v18, current

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