Real Line Intervals as Closed, Bounded Non-Compact Spaces

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Discussion Overview

The discussion revolves around the characterization of real line intervals, specifically whether every interval (a,b) can be classified as a closed and bounded non-compact topological space. The scope includes theoretical considerations of topology and metric spaces.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that every interval (a,b) is a closed and bounded non-compact topological space.
  • Another participant challenges this by stating that the interval (a,b) is not closed.
  • A different participant questions the definition of closed and non-compact, asking in what space these properties are being considered.
  • One participant clarifies that they meant a closed bounded metric space, indicating that the interval is closed in itself.
  • Another participant argues that boundedness is not a topological property, noting that one can introduce a new metric that maintains the same topology while making the space bounded, and points out that (a,b) is homeomorphic to the real line, which is not bounded.

Areas of Agreement / Disagreement

Participants express differing views on the properties of the interval (a,b), particularly regarding its closure and boundedness. There is no consensus on the classification of (a,b) as a closed and bounded non-compact space.

Contextual Notes

Participants highlight that boundedness may depend on the choice of metric and that the definitions of closed and bounded can vary based on the topological space considered.

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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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