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This is the definition from wikipedia

A sequence $X_1, X_2, \cdots $ of real-valued random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable X if $ \lim _{n\to \infty }F_{n}(x)=F(x) $, for every number $x \in \mathbb{R}$ at which F is continuous. Here $F_n$ and $F$ are the cumulative distribution functions of random variables $X_n$ and $X$, respectively.

I am confused by this definition because I can interpret it in two ways.

Let $E$ be the set of points on which $F(x)$ is continuous, then

$1)$ $\forall x \in E \ \forall \ \epsilon >0 \ \exists \ n $ s.t $|F_N(x)-F(x)|< \epsilon \ \ \forall N \ge n$

$2)$ $ \forall \ \epsilon >0 \ \exists \ n$ s.t $|F_N(x)-F(x)|< \epsilon \ \ \forall x \in E \ \ \forall N \ge n $

Which of the above correspond to convergence in distribution?

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The first one is what is meant. The latter of the two statements you mentioned would mean the convergence happens uniformly, i.e. there is a point at which the differences between the function and the limit is less than $\varepsilon$ for all $x$ simultaneously. This is much stronger than the first one, and usually statements like these are interpreted in the 'weak sense' (i.e. 1) unless otherwise indicated (in which case you would need to see the word 'uniform' somewhere).

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I would interpret the text as:

$$\forall x: (F\text{ is continuous at } x)\implies \lim_{n\to\infty} F_n(x)=F(x)$$

Or, with your definition of $E$, that would be

$$\forall x\in E: \lim_{n\to\infty} F_n(x)=F(x)$$

which is what your first option describes. You could also describe this by saying that $F_n$ converge pointwise to $F$.

The secon is more restrictive and implies that $F_n$ converge uniformly to $F$.

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