So I found an interesting example of a topology a while back but I can't remember the exact properties it had so I am hoping if I describe the topology someone can link me to the topology I am after. It was a short paper that I had found.
The basic idea was points in the plane were used to form triangles by using rational endpoints on the x-axis. That is two points on the x-axis would form a base then a point in the plane would be used to form the triangle.
If I recall the space was Hausdorff but I don't remember the other properties.
Best Answer
You’re probably thinking of Bing’s Sticky Foot Space, which is an example of a countable, connected Hausdorff space. I described it in this answer, and the picture here may also help.