Register to reply 
Topology of the unit interval 
Share this thread: 
#1
Jan3013, 01:07 PM

P: 29

Hi all,
I need help with something basic but I'm not sure how to handle it. The doubt is about how to consider the topology of the unit interval I=[0,1] inherited of the real line with its usual topology (intervals of the type (a,b)). I think that is just to pay attention to the definition, I mean, the open subsets of 'I' would be the intersection of a usual open interval and 'I'. In this way, 'I' itself would be a open subset of the inherited topology, and all the sets of the form [0,x), (a,b) and (y,1] with 0 < x,a,b,y <1  would be open sets of the inherited topology. Please, can anyone tell me if I'm right? Thanks in advance. 


#2
Jan3013, 02:06 PM

P: 810

Sounds about right.
Note that some sets in the subspace are open sets even if they aren't open in the larger space. For instance, [0, 1] is closed in R. But when we consider [0, 1] as a subspace, it's open (because the entire topological space is required to be open in any topology). Similarly, [0, 1), which is neither open nor closed in R is open in [0, 1]. 


#3
Jan3013, 02:17 PM

P: 29

Thank you TacTics



Register to reply 
Related Discussions  
Basic topology proof of closed interval in R  Calculus & Beyond Homework  1  
Unit tangent, unit normal, unit binormal, curvature  Calculus & Beyond Homework  4  
Should I do a unit in topology?  Academic Guidance  2  
The usual topology is the smallest topology containing the upper and lower topology  Topology and Analysis  2  
The usual topology is the smallest topology containing the upper and lower topology  Calculus & Beyond Homework  0 