Topology is a form of geometry where we can stretch or compress space as much as we want, but if we cut it we must put it back together. This gives it a nickname of "rubber sheet geometry". You may think that not many interesting things can be done, but you would be wrong. For a start, any two-sided solid (has an inside and outside) is a sphere, a torus (doughnut-shaped), a torus with two holes, a torus with three holes, and so on. But what about other shapes? In fact, it doesn't matter about the location of the holes. Firstly, only two-sided holes count. Likewise, a Y-shaped hole through a sphere is actually a two-holed torus! First, the part between two arms of the Y is stretched until it comes out of the other, resulting in a V-shaped hole. These can be separated to form a two-holed torus.
Topology is also about knots, but here you must also put the surrounding space back together. This is important as otherwise you could cut any knot, untie it, and put it back in a circle. For example, Alexander the Great cut the Gordian knot. After taking the cut pieces of rope, he could have put it back to make a circle if he wanted to. However, you cannot glue back the entire surrounding space.
Topology can be done in many dimensions, or even fewer than 3. For example, a line [0,1], which is closed, can be mapped to the line [0,2] by doubling each point's co-ordinate. However, it cannot be mapped to an infinite open line. The line (0,1), which is open, can be mapped in such a way. This is as there is no highest or lowest point on either line, as there is no highest real number less than 1. In these lines, the distance between one point and another is countable. However, there is a line called the long line or Alexandroff line. To visualise it, imagine a square divided into uncountably many horizontal lines. You start at the top, working your way down once you reach the end of a line. The distance between any two points in separate horizontal lines is always uncountable. This is an open line that cannot be mapped continuously to the real number line, despite containing the same number of points.
Related entries Möbius band
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