00:01
For this problem, we are told that three types of classic topological surfaces are shown below.
00:05
We have that the sphere and taurus have both an inside and an outside, and we are asked, does the klein bottle have both an inside and an outside? we are asked to explain.
00:15
Now, one way of understanding having or something having an inside and an outside is to consider, for instance, with the sphere.
00:24
I'll just look at a cross -section of our sphere.
00:27
We can think of having...
00:30
Points pointing in the direction of outside, pointing away.
00:38
We can imagine that we have a little, almost like a little needle that we're sticking up out of the sphere and we can move it along the outside.
00:46
Alternatively, we can also imagine a needle that we've poked into the sphere and we move that around.
00:51
You can see that it's pointing in the opposite direction, and there's no way to smoothly transition a point or a vector pointing outwards to arrive at a vector pointing inwards.
01:04
Same deal with the taurus...