Real Line Intervals as Closed, Bounded Non-Compact Spaces

[symbol0]

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

Is this correct?

 

[g_edgar]

Interval (a,b) is not closed.

 

[Preno]

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.

 

[symbol0]

Yes, that's what I meant: a closed bounded metric space (closed in itself).
Thanks for clarifying this for me.

 

[tim_lou]

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).

 

[symbol0]

Thanks tim_lou