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    The topological notion of a neighborhood, necessary to de... — Carmelics
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    Challenges→In a continuous manifold, the concept of the manifold and its continuity properties can be separated from its metrical structure.

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

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

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

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

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

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    Reasons Against

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

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

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

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    Related

    Any intuitive notion of 'nearness' in neighborhoods relies on unstated distance ...General topology's abstraction cannot fully escape the metric intuitions that hi...In a continuous manifold, the concept of the manifold and its continuity propert...Non-metrizable spaces (e.g., cofinite topology) have well-defined neighborhoods ...
    +3 moreShow less
    Open sets can be defined purely axiomatically without any proximity concept, mak...Pedagogically, neighborhoods are taught via metric examples, suggesting metric s...That intuitions originate metrically doesn't mean formal definitions require met...

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    claim
    Perspectives
    2 (1 for, 1 against)
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