Showing that a half-open set is neither open nor closed

[Mr Davis 97]

Homework Statement

Where ##a,b\in \mathbb{R}##, show that ##[a,b)## is not open.

Homework Equations

The Attempt at a Solution

I need to show that there exists an ##x\in [a,b)## such that for all ##\epsilon > 0##, ##B_\epsilon (x) \not \subseteq [a,b)##. To this end put ##x=a##, and let ##\epsilon > 0##. Then ##B_\epsilon (x)= (a-\epsilon, a+\epsilon)##, and since ##a-\epsilon < a##, we have that ##B_\epsilon (x) \not \subseteq [a,b)##.

Is noting that ##a-\epsilon < a## enough to prove that one is not a subset of the other?

 

[fresh_42]

Yes.

And why isn't it closed?

 

[WWGD]

Just a comment. I assume you may have meant half-open interval, not half open set since I can't think of any other setting in which a set may be half open.

 

[fresh_42]

Just a comment. I assume you may have meant half-open interval, not half open set since I can't think of any other setting in which a set may be half open.

The set of molecules forming your front door

 

[WWGD]

Actually, I ruined someone's joke that a set is not like a door, open or closed. A door may be open , closed and locked or closed and unlocked. But I can't think of unlocked open sets ;).

 

[fresh_42]

But I can't think of unlocked open sets ;)

"Many of the perpetually open Denny’s restaurants were built without locks, which was problematic when they decided to close down for Christmas for the first time in 1988."