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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The topological notion of a neighborhood, necessary to define continuity, cannot be specified without some measure of proximity that smuggles in metrical commitments.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Open sets can be defined purely axiomatically without any proximity concept, making neighborhoods logically independent of metrics.
      ?

      Think about whether this reason is strong or weak

    • 2.Non-metrizable spaces (e.g., cofinite topology) have well-defined neighborhoods proving topology transcends metric commitments entirely.
      ?

      Think about whether this reason is strong or weak

    • 3.That intuitions originate metrically doesn't mean formal definitions require metric structure—conflates epistemology with mathematical necessity.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Any intuitive notion of 'nearness' in neighborhoods relies on unstated distance assumptions, even in purely set-theoretic definitions.
      ?

      Think about whether this reason is strong or weak

    • 2.General topology's abstraction cannot fully escape the metric intuitions that historically motivated and validated its core concepts.
      ?

      Think about whether this reason is strong or weak

    • 3.Pedagogically, neighborhoods are taught via metric examples, suggesting metric structure is epistemically prior to topological generalization.
      ?

      Think about whether this reason is strong or weak

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