Understanding the Concept

Let’s imagine an organism🐛 that lives attached to a two-dimensional sphere \( S^2 \) (what we normally think of as a sphere🌐). In other words, to that organism, the sphere is the universe itself.

Let’s call that organism \( A \).

Loops on the Sphere

Now, consider any loop⭕️ (a closed curve that does not intersect with itself) on the sphere that surrounds \( A \), and fix a point \( P \) on the loop. \( A \) is inside the loop, but if we keep point \( P \) fixed and continuously deform the loop, expanding it more and more, \( A \) will soon be outside the loop. From this, it can be thought that \( A \), which is inside the loop on the sphere, is also outside the loop. The same is true in reverse.

Here, we used the terms “inside” and “outside” of the loop, but if we consider the loop to be the equator, the question arises: is \( A \) inside or outside of the loop? Instead, understand that a loop simply divides the sphere into two domains.

Homotop (or Homotopic)

Next, if we keep point \( P \) fixed and continuously deform the loop, it can be shrunk down to point \( P \). In this case, we say that the loop and point \( P \) are homotop (or homotopic) to each other.

In general, a domain in which such a property holds is said to be simply connected.

Example

Incidentally, the torus🍩 is not simply connected.