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In the definition of a manifold, one defines, in particular, a chart as a homeomorphism $x:U\to O$ where $U\subseteq M$ is an open set of the topological space $M$ and $O\subseteq \mathbb{R}^n$.

Question : Is the empty function $x:\emptyset\to\emptyset$ a chart ?

I think the answer is vacuously yes.

Guest
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    For that definition of chart, the answer is yes. But not vacuously. That is not what «vacuously» means when applied to definitions. – Mariano Suárez-Álvarez Jan 21 '15 at 19:28
  • @MarianoSuárez-Alvarez Well I thought the function would be injective since the condition $\forall x,y\in\emptyset:x\neq y\implies f(x)\neq f(y)$ is vacuously true (that is, if it were false, we could exhibit a counter-example, which we can't do here). Same thing for surjectivity : $\forall y\in\emptyset,\exists x\in\emptyset:f(x)=y$ is vacuously true for me. I thought the inverse function of $x$ would be $x$. So do you mean that the continuity of $x$ is not vacuously true ? That is, there is an (unique) open set of $\emptyset$ which is $\emptyset$ and $x(\emptyset)=\emptyset$ ? – Guest Jan 21 '15 at 19:38
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    Yes , one says that it is vacuously true that the function is injective and surjective, but I don't think anyone would say that it is vacuously true that it is a chart. (I would not say that it is vacuously true that it is an homeo, for example) – Mariano Suárez-Álvarez Jan 21 '15 at 19:40

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