# Bounded topological space?

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Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?

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.