Lower Limit Topology Clopen Sets

[tylerc1991]

Homework Statement

Let T be the lower-limit topology on R. Is (R,T) connected? Prove your answer.

The Attempt at a Solution

Since there exists a proper subset V of R such that V is both open and closed (since all intervals of the form [a,b) are open and closed), then (R,T) is disconnected.

Well, I think I need to provide some justification of why the intervals of the form [a,b) are open and closed: [a,b) is open because it is an element of the basis for the lower limit topology. However, I am not exactly sure of how to go about showing that [a,b) is closed. I was trying to show that it's complement is open, but this didn't exactly get me anywhere. The complement of [a,b) is (-infinity,a) union [b,infinity). I can maybe grasp how [b,infinity) is open because it has the form of an open set in the lower limit topology, but what about (-infinity,a)? Is this open?

Anyway, the question essentially comes down to showing that [a,b) is closed in the lower limit topology. Thank you for any help!

 

[Dick]

Show (-infinity,a) and [b,infinity) are infinite unions of basis sets.

 

[tylerc1991]

Show (-infinity,a) and [b,infinity) are infinite unions of basis sets.

(-infinity,a) = U_(n in Z) [a-(n+1),a-n) where U - union, and Z is the set of non-negative integers

[b,infinity) = U_(n in Z) [b+n,b+(n+1)) where U - union, and Z is the set of non-negative integers.

 

[Dick]

(-infinity,a) = U_(n in Z) [a-(n+1),a-n) where U - union, and Z is the set of non-negative integers

[b,infinity) = U_(n in Z) [b+n,b+(n+1)) where U - union, and Z is the set of non-negative integers.

Sure.