Definition of product implies the existence of continuous function?

Question

What is the definition of product that it's being considered here to have the existence of a unique continuous function ?

Let $X_i$ be path connected topological spaces.

Then $\prod_i X_i$ is path connected.

Proof.

Assume $X=\prod_{i\in I}X_i$ with $X_i$ path-connected. Let $x=(x_i)_{i\in I}$, $y=(y_i)_{i\in I}$ be two points in $X$. By assumption, thee exist continuous paths $\gamma_i\colon[0,1]\to X_i$ with $\gamma_i(0)=x_i$ and $\gamma_i(1)=y_i$.

By definition of product, there exists a unique continuous $\gamma\colon[0,1]\to X$ such that $\pi_i\circ\gamma=\gamma_i$ for all $i\in I$ where $\pi_i$ is the projection $X\to X_i$. That makes $\gamma$ a path from $x$ to $y$.

This proof is from here Product of path connected spaces is path connected .

Answer 1

The product of topological spaces has as underlying set the Cartesian product: an element of $\prod_iX_i$ is a tuple $(x_i)_{i \in I}$. In other words, an element of $\prod_i X_i$ has a coordinate $x_i$ in each one of the $X_i$. The product topology is the one generated by sets of the form

$$\prod_{i} U_i,$$

where all the $U_i \subseteq X_i$ are open and $U_i = X_i$ for all but finitely many $i$'s. Hence, sets like the above form a basis for the product topology on $\prod_i X_i$.

Now, why does this definition has anything to do with continuity? For this, notice that for each $k \in I$ we have a very natural map, called the $k-$th projection:

$$ \pi_k \colon \prod_i X_i \to X_k, (x_i) \mapsto x_k.$$

Now you should check the following:

• For all $k \in I$, $\pi_k$ is continuous
• A map $\phi \colon Z \to \prod_i X_i$ is continuous if and only if all the compositions $\pi_k \circ \phi \colon Z \to X_k$ are continuous.

When you think your background is solid enough (and when you have checked the above by yourself), I encourage you to have a look at this very useful notes of professor Emily Riehl. They are not particularly difficult, but they are a bit abstract, so that it is probably better to have already some confidence with the concepts treated in order to gain the most out of it.

Answer 2

Topological space $X$ equipped with continuous maps $\pi_i:X\to X_i$ for every $i\in I$ serves as product for the family $(X_i)_{i\in I}$ of topological spaces iff for every topological space $Y$ and every family of continuous functions $(g_i:Y\to Y_i)_{i\in I}$ there is a unique continuous function $g:Y\to X$ such that $g_i=\pi_i\circ g$ for every $i\in I$.