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I am learning homology. I cannot see in which sense the singular $p$-simplex is singular. It seems quite regular.

Any idea?

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    $\begingroup$ A singular simplex is by definition a continuous map of the (good old) simplex into your topological space. The image of such a thing could be extremely nasty, hence the name. (By way of contrast, in simplicial homology, we only think about nicely-embedded simplices.) $\endgroup$
    – Schemer
    Commented Aug 21, 2015 at 14:43
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    $\begingroup$ I wondered about this as well when I was first learning it, and realized that a better name would be "a (possibly) singular simplex", but that's a nightmare to write. An example of an actually singular simplex (which is also a "(possibly) singular simplex", of course) in the plane is the map $(s, t, u) \mapsto (0,0)$, where $(s, t, u)$ is a point of the standard 2-simplex ($ 0 \le s,t, y \le 1; s + t + u = 1$). That's certainly a singular map! $\endgroup$ Commented Aug 21, 2015 at 15:11

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