Is the Unit Interval [0,1] Open in Its Inherited Topology from the Real Line?

[dapias09]

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.

 

[Tac-Tics]

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]

Thank you Tac-Tics