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The following illustration is given to explain Van Kampen Theorem by the book from Hatcher.

enter image description here

In the above example, the line saying "points inside $S^2$ and not in $A$ can be pushed away from $A$ toward $S^2$ or the diameter...". This statement looks quite cryptic! What stops me from pushing all the points inside $S^2$ towards $S^2$. The only way I can visualize points mapping to the diameter is via imagining a cylinder along the diameter as in 2nd diagram, and then retracting it onto that diameter. But all points inside the sphere and outside the said cylinder will be retracted to the sphere. And here is the thing - those points, i.e. inside sphere and outside cylinder, are kind of random (no symmetry), if these points can be pushed (retracted) to the sphere, then why not every point inside the sphere.

Now my question is what is stopping me here in pushing all inside points to the sphere $S^2$.?

Your support would be greatly appreciated.

user3001408
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    Very briefly, what stops you from pushing all points inside $S^2$ toward $S^2$ is that the direction in which you push is not well-defined at the origin. The intention is that "pushing" is a brief description of a deformation retraction by straight line homotopy, and the direction of pushing must be well-defined and continuous. – Lee Mosher Jul 29 '17 at 14:45
  • You might recall, by the way, the theorem which says that the closed 3-ball $B^3$ does not deformation retract onto its boundary $S^2$. In fact, it does not even retract onto its boundary. A brief description of this theorem might be: you can't push the inside of $S^2$ onto $S^2$. You can therefore use this theorem as a guide to the meaning of the more intuitive language in this textbook regarding "pushing". – Lee Mosher Jul 29 '17 at 14:46

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As Mosher points out in the comments it is impossible to push all points in $S^2$ to the boundary $S^2$ as this would create a contradictory deformation retraction of $B^3$ to $S^2$.

It should also be noted even if one could do this one wouldn't have motivation to do so in this problem. Here we want points in $S^2$ not in $A$ as we are considering the complement of $A$. And the lack of deformation retraction as per Mosher leads us in the direction that the diameter will be included.

Birch Bryant
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