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The whole question is all in the title.

Let $X$ be a topological or metric space, $E\subseteq X$.

Has anyone invented a symbol for the fact that "$E$ is open in $X$"?

(especially, if $X$ is a metric space with metric $d$, is there a symbol to represent this?)

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    A topological space $X$ is given together with a topology $\tau_X$. The way to say $E$ is open in $X$ is just $E\in\tau_X.$ – 2017-02-01
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    @mathbeing yap, I am just aware of that, so I edited the post soon. – 2017-02-01
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    I think not. For the record, mathematics is not about inventing a symbol for every possible notion; very often using actual words is useful. But if you _need_ such a notation for some reason, make it up yourself. – 2017-02-01
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    $(X,d)$ metric space with $\overset{\circ}{E}=E\subseteq X$. Here $int(E)=\overset{\circ}{E}$ is the interior. – 2017-02-01
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    It seems weird to make up a word, when almost everyone would do what mathbeing said. – 2017-02-01
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    @DanielBernoulli Yes, it is indeed a notation for that. But I feel there should be a more direct symbol for this($E^\circ=E$ is more likely an equivalent statement). – 2017-02-01
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    I have seen instructors write $E^{\rm open} \subset X$, $F^{\rm closed} \subset X$, and $K^{\rm compact} \subset X$ with their obvious meanings, and this could be extended to just about anything like $Y^{\rm uncountable,\ nowhere\ dense,\ bounded} \subset X$. This was just blackboard usage, not formally done in print. – 2017-02-01
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    In class I often use what @UmbertoP. suggests, although I tend to place the word right on top of the $\subset$ symbol, to represent a restriction of the relation $\subset$. – 2017-02-01
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    Given your edit, you could just say "let $\tau$ be the metric topology on $X$ generated by $d$, and let $E\in\tau$". – 2017-02-01

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Some authors use $U \subset_{\circ} X$ as a notation for: $U$ is an open subset of $X$.

For instance, see some of the lecture notes of professor Bruce Driver.