A topology playground. Everything here is made of magic clay.
The one rule
Stretch yes. Cut no. Glue no.
Imagine every shape is a lump of clay that can bend, stretch, squish and twist as much as you want. Two shapes are the same in topology if you can turn one into the other without tearing a new hole and without sticking two parts together.
So topology does not care how big something is, or whether it is round or pointy. It only cares about what is connected to what, and how many holes there are.
Experiment 1
Turn the coffee mug into a donut
Drag the slider slowly. Watch the hole. It never closes and no new hole ever appears.
Coffee mug
SquishHoles: 1
Why they are the same: a mug only has one hole, and it is the handle. The part you pour coffee into is not a hole. It is a dent. A dent goes in and comes back out the same way, so you can just push it flat with your thumb. No tearing needed. The handle hole goes all the way through, so it has to survive. A donut also has exactly one hole. One hole equals one hole, so mug equals donut.
Experiment 2
Now try it with a ball
Squish the ball as hard as you like. Pancake it. Sausage it. Then check the hole counter.
A perfectly normal ball
SquishHoles: 0
Stuck at zero. Squishing can change the shape as much as it likes, but it can never change the number of holes. To get from 0 holes to 1 hole you must break the clay. That is cutting, and cutting is against the rule. So a ball and a donut are different, forever.
Experiment 3
The detective trick: take away one point
This is the trick that proves a line and a pancake are different shapes, which is exactly what your quiz question was about. Pull one single point out of each and count the pieces left over.
A piece of string (1D)
Pieces: 1
A pancake (2D)
Pieces: 1
Here is the punchline. Squishing never changes how many pieces something is in. So if string and pancake were secretly the same shape, removing one point would have to do the same thing to both. It does not: the string falls into 2 pieces, the pancake stays in 1 piece because you can walk around the tiny hole. Different result means different shape. A 1D thing can never be topologically the same as a 2D thing.
Experiment 4
Hole counting championship
In topology, the number of holes is the object's secret ID card. Count carefully. A dent is not a hole.