Real Line Intervals as Closed, Bounded Non-Compact Spaces

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SUMMARY

Every interval (a,b) of the real line is not a closed and bounded non-compact topological space. While (a,b) is bounded as a metric space with the usual metric, it is not closed in the standard topology of the real line. The discussion clarifies that boundedness is not a topological property, as any metrizable space can be given a new metric that alters its boundedness without changing the topology. Specifically, (a,b) is homeomorphic to the real line, which is not bounded.

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I had this thought:
Every interval (a,b) of the real line is a closed and bounded non-compact topological space.

Is this correct?
 
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Interval (a,b) is not closed.
 
symbol0 said:
I had this thought:
Every interval (a,b) of the real line is a closed and bounded non-compact topological space.

Is this correct?
Closed and non-compact in what space? Trivially, any topological space is closed in itself (which is why we don't really say that something is a "closed topological space"), is this what you mean?

I don't know what a bounded topological space is, but it is bounded as a metric space with the usual metric.
 
Last edited:
Yes, that's what I meant: a closed bounded metric space (closed in itself).
Thanks for clarifying this for me.
 
boundedness isn't really a topological property, for any metrizable space, one can introduce a new metric:

d' = min{d, 1}

and the resulting topology will be the same and the metric will be bounded. More obviously and specifically to your example, (a,b) is homeomorphic to the real line, which isn't bounded. (hence, boundedness isn't really a topological property).
 
Thanks tim_lou
 

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