Networks and maps on the Plane, the Sphere , and the Torus: See activity for networks on the torus.
"Bounded" In the plane- A line segment is bounded. A line is not bounded. A circle is bounded. A parabola is not bounded. "Bounded" means the object can be visualized in a box.
"closed" A line segment with endpoints is closed. A circle is closed. A line is closed. A line segment missing one or both endpoints in not closed. A circle missing one point is not closed.
"open" A line segment missing both endpoints is open. A line is open. A circle missing one point is open.
"connected" A line, a line segment, a circle, a circle missing one point, and a parabola are all connected.
A line segment missing an interior point is not connected.
It has two pieces.
The following two letters are topologically
The following two letters are
topologically equivalent to each other but not to the previous two letters. This can be seen by removing a single point from these letters
which will not disconnect the curves, as it does with the previous letters.
The letter T missing the point where the top meets the vertical line segment is not connected. It has three pieces.
The letter Y missing the point where the top meets the vertical line segment also is not connected. It has three pieces.
T and Y are topologically the same (equivalent)! this can be seen by bending the tops to the letters up or down and stretching and shrinking the lengths as well.
Questions: Is a triangle
topologically related to another triangle?
Yes. Stretch the sides and you'll also transform the angles.
Is an circle topologically
related to a line?
The circle is bounded and still connected when you remove a point.
The line is not bounded and is disconnected when you remove a point.
Is a triangle topologically related to a square?a circle? yes.