The judgment is simple. Make any closed curve in the area, and then use the geometry.

From the point of view, whether the interior of this closed curve contains points that are not in the region.

If so, it is not simply connected. If the interior of any closed curve does not contain

A point in a nonregion is simply connected.

For example, on the right, make a unit circle, which contains points whose origin is not in the region.

So it's not simply connected. On the left side, there must be no closed curve.

A point on the negative real axis, so it looks simply connected.