Find the Closure of these subsets

[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.

[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.

[pivoxa15]

My guesses are displayed in the OP.

[matt grime]

And what are your justifications for them?

[HallsofIvy]

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?