set X = {a,b,c,d}
These are the three collections of subsets:
1 = { empty set, X, {a}, {b}, {a,b}, {a,c,d}, {b,c,d}, {c,d} }
2 = { empty set, X, {a,b}, {c,d} }
3 = { empty set, X, {a,b}, {b}, {b,c,d} }
Which of 1,2 and 3 are Hausdorff and why?
PS if there is any ambiguity in what I've written, then just notify me and I will try to clarify.
I am new to the definition of Hausdorff, but when I look at it, I would try to find open sets which include the components of these subsets, see if their intersections equal the empty set.
However, I am not sure which open sets I can use?
For example, is it too simplistic to say that 1 is not Hausdorff, because open set {a} intersecting with open set {a,c,d} = {a} which does not equal the empty set?
Therefore, since all three collections are topologies, the subsets in each case 1,2 & 3 are called the open subsets of X.
But none of these can be Hausdorff since any distinct pair of points from X, e.g. a & b contained in any of the sets in 1,2 or 3, intersected with X will not be empty...is that a correct viewpoint?