Topology of the unit interval


by dapias09
Tags: interval, topology, unit
dapias09
dapias09 is offline
#1
Jan30-13, 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.
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
Tac-Tics
Tac-Tics is offline
#2
Jan30-13, 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].
dapias09
dapias09 is offline
#3
Jan30-13, 02:17 PM
P: 29
Thank you Tac-Tics


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