Find the Closure of these subsets

1. Sep 19, 2007

pivoxa15

1. The problem statement, all variables and given/known data
X=R real numbers, U in T, the topology <=> U is a subset of R and for each s in U there is a t>s such that [s,t) is a subset of U where [s,t) = {x in R; s<=x<t}

Find the closure of each of the subsets of X:

(a,b), [a,b), (a,b], [a,b]

3. The attempt at a solution
I don't understand the topology of X very well. So have trouble finding the closure. No metric is allowed?

I used complements to work out open and closed properties and came to the conclusion :
[a,b), [a,b), [a,b] and [a,b] as the closures of the above respectively.

Last edited: Sep 19, 2007
2. Sep 19, 2007

matt grime

You seem to have an obsession with metrics. Stop it. Firstly, try to work out what a typical open set looks like, thus you know what a typical closed set looks like. For instance, is (0,1) open? Is it closed? (It can of course be both.) What about (-inf,0], (-inf,0), (0,inf), or [0,inf)?

And what's stopping you making a sensible guess? I mean it is surely the case that the closure of (a,b) is going to be one of (a,b), (a,b], [a,b) or [a,b], so you have to see which of those is closed, i.e. which has open complement.

Last edited: Sep 19, 2007
3. Sep 19, 2007

pivoxa15

My guesses are displayed in the OP.

4. Sep 20, 2007

matt grime

And what are your justifications for them?

5. Sep 20, 2007

HallsofIvy

Staff Emeritus
Do you remember back in Calculus I (or maybe Precalculus) when you worked with "closed intervals" and "open intervals"? Those names come from this. Is (a, b) a closed or open interval? What can you do to make it closed?