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Consider the following:

The Grothendieck topology is not a topology.

A Riemannian metric is not a metric.

The notation $\lim_{\to}$ is a co-limit, not a limit.

(Two maybe different examples)

The complex plane is (for algebraic geometers) a curve. Even worse, the Riemann sphere is also a curve.

An affine group scheme is a representable functor.

What are some mathematical objects that follow this unfortunate pattern?

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    $\begingroup$ This could be of interest : math.stackexchange.com/questions/747012/… $\endgroup$ – Arnaud D. Dec 12 '16 at 21:24
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    $\begingroup$ I don't really understand how something like "the Riemann metric is not a metric" works given that metric may have many different senses throughout mathematics. But anyway, the field of one element is not a field. $\endgroup$ – rschwieb Dec 12 '16 at 21:35
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Elliptic curves are not related to ellipses, but to elliptic integrals.

Additionally, the monster group is not a monster, it's just misunderstood.

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From Wikipedia:

A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed.

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The inverse image is not necessarily the (direct) image of any function.

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A manifold with boundary is not necessarily a manifold (and when it is, it's arguably not "with boundary").

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I'm not sure whether this qualifies, since "series" does not have another meaning in group theory so far as I'm aware, but it is annoying given the potential for confusion between sequences and series in elementary analysis.

Let $G$ be a group. A (subgroup) series for $G$ is actually a sequence of subgroups $H_{1},H_{2},\ldots,H_{n}$ of $G$ satisfying $$1<H_{1}<H_{2}<\cdots<H_{n}=G.$$

Of course, this extends to any particular kind of series, e.g., normal series, composition series and chief series.

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The Peano-Jordan measure isn't a measure (the measurable sets don't form a $\sigma$-algebra), because of this reason some authors prefer the term "jordan content"

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