As my topic says I'm in search of a bit of help with some topology. This is some work for a graduate course in advanced topology and I'm having a bit of trouble understanding/proving a couple problems. I realize this is an advanced topic so it may not be understood by many so I will give a sample problem of something I'm having trouble with and we can go from there if anyone can help I would really appreciate it, thanks!


problem 1. Let X be R union {x}. Let T={U such that U is open in R} union {x}.
1a) is T a topology on X
1b) show X has the property that every sequence has a limit point.
idea: I believe this works because every open set in X looks like (a,b) union {x} thus {x} is a limit point of every open set in X, thus every sequence in X has a limit point which happens to be {x}
1c) show X is non-hausdorff
idea: I believe X is non-hausdorff because for any two open sets in X, say (a,b) union {x} and (c,d) union {x} these open sets always have the point {x} in common so no two open sets in X can be disjoint.


Above I have listed one of the problems I am currently working on and a little idea for what I'm thinking thus far (Which I'm not sure on yet).