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On the Wikipedia page of "product topology", they define the Cartesian product of the topological spaces $Xi$ as $X= \Pi_{i\in I} X_i$

Now, in class, my professor defined $\Pi_{i\in I} X_i = \{x_i: I \rightarrow \cup_{i \in I} X_i, x_i \in X_i, \forall i \in I\}$

Now I am trying to understand that last equality given my professor. I don't quite understand it and I don't understand what it bring to the original definition of the cartesan product of sets.

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    I think your definitions are a bit "Rocky". Read https://en.wikipedia.org/wiki/Product_topology#Definition carefully. – 2017-02-05
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    The notation you're using is confusing: $x_i $ is a function defined on I$, and $x_i$ is a point later? See the correction in my answer. – 2017-02-05

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It's the standard set theory definition which makes sense when we assume the axiom of choice: the product consists of all choice functions.

So I would write

$$\prod_i X_i = \{f: I \rightarrow \cup_i X_i: \forall i: f(i) \in X_i\}$$

We can also write $(f(i))_i$ for this function or $(p_i)_{i \in I}$ where every $p_i = f(i) \in X_i$. There you can see the similarity with the more familiar (probably) finite Cartesian product: we can also identify $X \times Y = \{(x,y): x \in X, y \in Y$ with the set of functions $f$ on $\{0,1\}$ into $X \cup Y$ that obey $f(0) \in X, f(1) \in Y$. The function notation makes the intuitive idea of "infinite tuples" more precise, and embeds it into familar set theory.