I've looked through quite a few resources on the internet and I can't really find a good answer to this question. I am just starting to get into topology and manifolds and I came across the statement "A coffee mug and a torus are topologically the same....any closed loop and a circle are topologically the same."
I mean obviously, they are geometrically pretty different. So what are the properties that topology is specifically concerned with? I know manifolds are a complete different story and give us much greater insight into the geometry of our given shape, but I'm specifically wondering what topology "measures." A mathematical answer and a layman answer would be greatly appreciated, thanks!
(if this is too general of a question, maybe give the 3 most important properties)