It's a matter of people imputing notions of "special" where they don't belong. Hence the importance of "proof" in math.<p>ETA: People are surprised that they can turn a sweater inside-out thru a sleeve or neck hole ONLY because they've imputed a "special" ability to the largest hole in the garment. The mathematical concept of "proof" strips away such imputations, leaving surprisingly unsurprising results - in this case, you can reverse a garment by pulling it thru one of its holes, be it the largest, smallest, or even a tear, because they're all just holes with nothing inherently topologically special about them. In a larger social concept: people tend to impute special attributes to various things where such attribution is not warranted; people who understand the concept of mathematical proof are less likely to get caught up in such incorrect imputation.
Love this question and all the answers. I find topology fascinating even though I understand maybe 5% of it. I took a decent amount of math as part of my CS degree, but beyond basic calculus it was concentrated in probability, stats, and linear algebra; never came near topology. In hindsight I wish I had taken more math, but as a 19-20 year old student at the time, I was happy to be done with it.<p>If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.
a fun introduction to topology is this article "Using Asteroids [the game] to explain the topological classification of 2-manifolds"<p><a href="http://everything2.com/user/sockpuppet/writeups/Using+Asteroids+to+explain+the+topological+classification+of+2-manifolds" rel="nofollow">http://everything2.com/user/sockpuppet/writeups/Using+Astero...</a>
When I think of topology the first I do is remember that size means nothing in topology.<p>So the long sleeve of the shirt? Shrink it right down - you are left with just a hole, and no sleeve.<p>Then flatten out the curvature of the neck and other parts, and you are left with a flat sheet of clothing, with two holes in it. There is no inside or outside to this, meaning the two sides are interchangeable, and that's why you can turn the real shirt inside out.
Is this relevant?<p><a href="https://upload.wikimedia.org/wikipedia/commons/5/55/Tesseract.gif" rel="nofollow">https://upload.wikimedia.org/wikipedia/commons/5/55/Tesserac...</a>