What is the interior and closure of [ 2,7 ) ?

[math25]

Hi , can someone please help me with this problem...


Let T be the collection of all U subset R such that U is open using the usual
metric on R.Then (R; T ) is a topological space. The topology T could also be described as
all subsets U of R such that using the usual metric on R, R \ U is closed and
bounded.

What is the interior and closure of [2,7 ) ?

I think int [ 2,7 ) = empty set

and closure of [ 2, 7 ) = [2,7 ) ?

thanks

 

[pka]

Let T be the collection of all U subset R such that U is open using the usual
metric on R.Then (R; T ) is a topological space. The topology T could also be described as
all subsets U of R such that using the usual metric on R, R \ U is closed and bounded. What is the interior and closure of [2,7 ) ?

This question is really flawed. In the usual topology on \(\displaystyle \mathbb{R}\) the set (-\infty,0) is open but its complement is not bounded.

In the usual topology the closure of [2,7) is [2,7] and the interior is (2,7).

 

[math25]

thanks, I know that in usual topology closure is [2,7] and int(2,7)...I missed second sentence first time, does this make sense now?

Let T be the collection of all U subset R such that U is open using the usual
metric on R and for some real number a, b ( - infinity, a) is subset of U and (b, infinity) is subset of U.
Then (R; T ) is a topological space. The topology T could also be described as
all subsets U of R such that using the usual metric on R, R \ U is closed and bounded. What is the interior and closure of [2,7 ) ?

 

[pka]

thanks, I know that in usual topology closure is [2,7] and int(2,7)...I missed second sentence first time, does this make sense now? Let T be the collection of all U subset R such that U is open using the usual
metric on R and for some real number a, b ( - infinity, a) is subset of U and (b, infinity) is subset of U.
Then (R; T ) is a topological space. The topology T could also be described as
all subsets U of R such that using the usual metric on R, R \ U is closed and bounded. What is the interior and closure of [2,7 ) ?

The closure of [2,7) is [2,7]. Think about the complement of [2,7] is (\(\displaystyle -\infty\),2) \(\displaystyle \cup\) (7,\(\displaystyle \infty\))
Recall that the closure is the smallest closed set containing the set.

Your right about the interior. Can an open set be bounded?

 

[math25]

I got it, thanks a lot for all your help.