1. ## rational sequence topology

let x be the set of real numbers and for each irrational x we choose a sequence {x $i$} of rationals converging to it in the euclidean topology .the rational sequence topology is the defined by declaring each rational open, and selecting the sets U $n$(x)={x $i$}(n from 0 to infinity) U {x} as a basis for the irrational point x.

can anyone tell me the following properties regarding the above topological space?

- basis
-sub basis
-interior and closure
-Both Open and Closed Sets
-Hausdorff Space and Metrizable Topology
-Relative Topology
-Finer and Coarser Topology
-Homeomorphism
-Connectedness
-Compactness
-Countability

2. ## Re: rational sequence topology

Eleven questions in your first ever post? That's got to be some kind of record!

Seriously, though: if you want to get a helpful reply, you should tell us what you've attempted already and where you're stuck, and possibly focus on one question at a time. (DON'T break it up into 11 separate topics and post them all at once!)

And if you haven't attempted anything yet because you don't know where to start, then re-read the chapter, asking yourself as you read each theorem and definition whether you can apply it to your current problem.