Why is an element in the countable product of real numbers regarded as a sequence?

Question

A sequence of real numbers is a just an enumerated list of real numbers. Now in my lecture notes it is mentioned that an element of the countable product topological space $\mathbb{R}^w$ (Tychonoff topology) can be regarded as a sequence of real numbers. I'm a bit confused by this statement. How can an element of the product topology be a real sequence? Could someone clarify this with a concrete example?

Answer 1

You probably think of $\Bbb R^\omega$ as the set of "$\omega$-tuples" (by analogy with $n$-tuples) with coordinates elements of $\Bbb R$. But then one has the sequence $(x_n)$, where $x_n:=\text{the n-th coordinate of x}$ for any element $x\in\Bbb R^\omega$.

But what's also kind of nice is the functional analysis flavor of considering $\Bbb R^\omega$ as the set of functions from $\Bbb N$ to $\Bbb R$. Then we simply have, for each $f\in\Bbb R^\omega$, the sequence $x_n:=f(n)$.