Is there a way to explain intuitively (i.e. without rigorous mathematical proofs) why unwinding a string wrapped around a pole is not a continuous mathematical transformation? He also states that a string wound round the pole once is topologically distinct from a string wound round twice, so then winding the string is also not a continuous transformation.
This is stated to be the case in 'An Introduction to Particle Physics and the Standard Model' by Robert Mann.
I can only guess at the full context, but here's what he may mean.
Assume the pole is infinitely long, and there's a loop of string that goes around the pole. Then there's no way to get the loop off the pole other than cutting it, moving it away and tying it back together. That's a discontinuous operation.
I think the term "unwinding" is confusing. It suggests that you have an end of the string to play with. The author may have used it because mathematicians refer to the number of times the string goes around the pole as the "winding number". (That number can be negative since the direction matters.)
Similarly, there's no way to take a string that goes around the pole twice and make it go around just once without cutting it.