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Bounded topological space?

  1. Mar 4, 2012 #1
    Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?
  2. jcsd
  3. Mar 4, 2012 #2
    Boundedness is not a topological concept. For example, take [itex]\mathbb{R}[/itex], then this is not bounded for the metric [itex]d(x,y)=|x-y|[/itex], but it is bounded for the metric [itex]d(x,y)=|atan(x)-atan(y)|[/itex]. However, the two spaces are homeomorphic.

    So it's possible that two metric spaces carry a homeomorphic topology, but that one is bounded and the other is not.

    This is why boundedness is not studied in topology.
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