Is Boundedness Applicable to Topological Spaces?

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SUMMARY

Boundedness is not a concept applicable to topological spaces; it is specific to metric spaces. For instance, the real numbers \mathbb{R} are unbounded under the metric d(x,y)=|x-y| but bounded under the metric d(x,y)=|atan(x)-atan(y)|. Despite these differences, both metrics can yield homeomorphic topologies. Consequently, the study of boundedness is irrelevant in the context of topology.

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Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?
 
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Boundedness is not a topological concept. For example, take \mathbb{R}, then this is not bounded for the metric d(x,y)=|x-y|, but it is bounded for the metric d(x,y)=|atan(x)-atan(y)|. 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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