Bounded topological space?

  • #1
blahblah8724
32
0
Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?
 

Answers and Replies

  • #2
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,314
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.
 

Suggested for: Bounded topological space?

  • Last Post
Replies
3
Views
343
  • Last Post
Replies
18
Views
608
Replies
5
Views
1K
Replies
6
Views
638
  • Last Post
Replies
5
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
17
Views
1K
Replies
14
Views
969
Top