An example of something that is not a subcomplex.

Question

I was reading this part in Allen Hatcher book :

But I did not understand it, can someone explain it to me with drawing please?

Edit:

I need a picture for the last 3 lines.

Answer 1

Here is the circle with its minimal cell complex structure:

Let's call the marked point $(-1,0)$. The only subcomplexes of this are the point $(-1,0)$, the whole circle, and the empty set. So for example take the map $S^1 \to S^1$ that sends $(x,y)$ to $(-|x|, y)$ (you can think of this as first projecting the circle to the vertical line segment from $(0,-1)$ to $(0, 1)$, and then "wrapping" that vertical line segment onto the left half to the circle). The image of this map is the left half of the circle, and that is not a subcomplex. If you glued a 2-cell using this as the attaching map, then the closure of that 2-cell would not contain the entire 1-cell, just part of it, and so it would not be a subcomplex.