# Bounded topological space?

1. Mar 4, 2012

### blahblah8724

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

2. Mar 4, 2012

### micromass

Staff Emeritus
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.