There is a smooth 1-manifold (a smooth curve of infinite arc length) embedded in a 2 dimensional euclidean space. This curve (of infinite arc length) is such that, there is one and only one point $P$ in the 2-d space such that any $\epsilon$ (arbitrarily small) neighbourhood of this point contains the entire curve except some parts of curve whose total arc length is finite. basically the arc length of the curve outside the neighbourhood is finite. I guess there certainly must be some notion in math that describes this type of geometry/topology or this type of embedding. I'd like to know where I can find such things. I am not interested in any other properties of the curve, except this situation. I'd like to know where I could find such a notion. Also if appropriate, I'd like to know similar things for higher dimensions such as a 'surface' embedded like this in a 4-d euclidean space.
I know almost nothing about topics such as topology, geometry or related topics, so please give some advice about where to look for such notions with terminology that I probably can understand in some intuitive sense. Also my tags could be inappropriate as I don't know much about it.