C-space, or configuration space. One dot on the map is one whole robot pose.
The big idea
Stop drawing the robot. Draw a dot instead.
A robot arm is a complicated thing with elbows and metal. But if you know its joint angles, you know everything about where it is. So instead of drawing the whole arm, you can just plot those angles as a single dot on a map.
That map is called configuration space, or C-space for short. Left side of every panel below: the real robot. Right side: the same robot, squashed down to one dot.
Panel 1
One joint. The map is a circle, not a line.
Spin the joint all the way around. Watch what happens when the angle passes 360.
The real robot
Its C-space
joint30°
Zero and 360 are the same place. The arm at 359 degrees and the arm at 1 degree are practically touching, so their dots must be neighbours on the map. On a straight line they would be at opposite ends, which is a lie. On a circle they are right next to each other, which is the truth. So the C-space of one spinning joint is a circle. Topology decided that, not you.
Panel 2
Two joints. The map is a donut.
Now there are two angles, so the map needs two directions: left-right for the shoulder, up-down for the elbow. That gives a square. But each direction wraps around, so the left edge is glued to the right edge and the top edge is glued to the bottom edge. Glue both and you get a donut.
The real robot
Its C-space (a flattened donut)
shoulder30°
elbow40°
no collision
the armyou are hereposes that crash into the obstaclethe long way through the middlethe short way through the wrap
Why the arrows on the edges matter. When your dot walks off the right side of the map it comes straight back in on the left side, because those two edges are the same place. Same for top and bottom. A robot programmer who forgets this makes the arm swing 350 degrees the long way instead of 10 degrees the short way. That is not a maths detail. That is a real arm smacking a real table.
Turn on the obstacle and look at the grey blob. One little obstacle in the real world becomes a big weird stain on the map, because many different poses crash into it. Planning a motion now means finding a path on the map that never touches the blob. That is what a motion planner actually does. It does not think about arms. It walks a dot around a stain on a donut.
Panel 3
So why did the course start with topology?
Because the shape of that map is the whole ballgame, and the shape is a topology question.
One spinning joint gives a circle. Two give a donut. Six give a 6-dimensional donut. A wheeled robot on a floor gives flat space times a circle, because it has a position and a heading. A drone or a hand in 3D gives something even stranger.
None of those are flat space. If you pretend they are flat, your robot takes the long way around, your interpolation jumps, or your orientation maths locks up. The famous gimbal lock problem in 3D rotation is exactly this: someone tried to describe a curved space with three flat numbers, and topology does not allow it. That is why 3D code uses quaternions instead.
The one sentence version. Topology tells you what shape the robot's map is. Everything after that, planning, control, collision checking, learning, is just moving a dot around on that map. Start with the wrong shape and everything built on top of it is wrong.