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
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
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