Topology is a branch of mathematics concerned with how things are connected to each other. It was invented by the mathematician Leonhard Euler when he tried to solve the problem of the Königsberg Bridges.
As a teenager, I found a wonderful introduction to topology in our local library - it was probably intended for children and it gave me a good grounding in the subject. Years later, I bought a more advanced book on the subject, obviously intended as a short University course. It was completely different! I fought my way through symbols, continuity and disjointed sets as far as the end of Chapter 1, but couldn't see much connection between it and the topology I had learned as a teenager.
I think the problem is that the strict mathematical approach is to define what you're talking about in as general a way as possible so that you don't restrict the solution by, for example, confining it to a limited number of dimensions, but to be absolutely rigorous about it. This makes the basics very tough going. The books-for-teenagers approach is to provide lots of examples and let the reader get a feel for the subject, which is the way that normal people learn things.
So now I'm going to try again and summarise here what I find.
R and S
R is the set of real numbers. It goes from 0 to positive infinity and in the other direction from 0 to negative infinity. R1 (normally done with a fancy hollow R) is the number line, a one-dimensional space with a point on it marked as the origin. Distance along the line is a real number, and every number in the set of real numbers R has a corresponding point on the number line R1. The number line is usually drawn as a straight line, but it doesn't need to be. It can wobble up and down like a sine wave, or spiral around like a helix. The only provisos are that distances are measured along the line from 0 and that it doesn't intersect itself, so each point has one unique distance from 0.
Are there any types of lines which are not equivalent to R1 ?
Well, there's a circle. Note that when we talk about a circle, we don't mean any of the points that are inside the circle - just the points along the perimeter1. If you mark a point on a circle as the origin, you'll find that distance from the origin increases along the line of the circle until you suddenly find yourself back at the origin again, something that never happens in R1. We can bend a circle into other shapes such as an ellipse, a square or the curve that forms the outside of a Pringle crisp. None of these change the fact that we still have a continuous line which we can measure distance against in one dimension (along the line) but which loops back on itself. The topological symbol for this shape is S1. S here stands for Sphere and this shape is sometimes called a 1-Sphere.
If we go to higher dimensions, we can see that R2 is a plane or infinite surface, while R3 is Euclidean space.
S2 is a sphere - that is, the surface of a ball. (Mathematicians use the term ball for the object which includes both the surface and the contents of the sphere.) Note that S2 is a two-dimensional surface even though it exists in 3 dimensions. Two numbers are enough to define the position of a point on the surface - for example latitude and longitude. One can imagine an ocean in the shape of a sphere (perhaps held in place by two concentric glass spheres of almost exactly the same radius). An intelligent amoeba swimming around in this ocean would not be aware of any third dimension, but could take measurements of the ocean and note that the laws of Euclidean plane geometry are not obeyed by the universe. Although locally there appear to be such a thing as parallel lines, in fact if you extend any pair of lines far enough in either direction they will meet.
So S2 has many of the properties of R2, being locally identical but differing on a grand scale.
We can extend both R and S into higher dimensions: R4 is a 4-dimensional space. S3 is a the 3-dimensional surface of a 4-dimensional sphere. S3 is a space. If you were in it, you could travel in any one of the six directions. But you'd eventually find you'd looped around to your original starting point.
Continuity is easy to understand but hard to define.
The real number line is continuous because you can move smoothly along it. There are no breaks. But how to define that?