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Mathsadness

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- TL;DR Summary
- This is not an assignment question but has been bugging me for a while. (0,1) is not compact as it has at least one open cover which does not have a finite sub cover. [0,1] is different as every open cover has a finite subcover for this.

For (0,1), the collection of neighborhoods N_e of q from (0,1) is an open cover. However, there exists e>0 such that it will not have a finite sub cover. Let us take e=0.5*min{|p-q|}, where p=/=q and both are from (0,1). I am not sure if the construction of e here is right, please correct me if not. Then every neighborhood will only cover one point from (0,1) which is q itself, hence a finite subcollection cannot cover (0,1) as there are infinitely many points. Then, similarly the collection neighborhoods for [0,1] is also an open cover given the neighborhood is taken from a point, say x from [0,1]. my question is then how can there be a finite sub covers for THIS particular open cover which consists of the collections of the neighborhood of x, where x is from [0,1]. By definition there has to be as the set is compact but I fail to see one for this particular open cover which is the collection of the neighborhoods of its points.

Thank you for any suggestion. i am new to this and would be obliged if the math shown is simplified.

Thank you for any suggestion. i am new to this and would be obliged if the math shown is simplified.