Projections of the Product Topology

Question

I've recently been studying product topologies from Schaum's General Topology and it appears that this is where topology begins to get fairly abstract (i.e. working in several dimensions). In the Product Topology chapter, the author defines

A projection is a function $\pi_{j_0}: X \to X_{j_0}$ where $X=\prod X_i$ is the product set and $X_{j_0}$ is a coordinate space. The function is given by $\pi_{j_0}(\langle a_i: i \in I\rangle)= a_{j_0}$

The author also goes on to say that the product topology is the topology generated by these projections.

I'd like to pose two questions:

1. Is it safe to say that a projection is merely a function that picks the $k^{th}$ element from an $n$-dimensional point in a product space and returns the value from $X_k$? e.g. let $\hat{x} \in \mathbb{R}^3$ with $\hat{x}= (4,5,6),$ then $\pi_2(\hat{x}) = 5$. Would that be sensible?

2. I'm having trouble understanding how the projections generate a topology. By "generate," surely we mean "take arbitrary unions of," but I am failing to see how $\displaystyle\bigcup \pi_i(\hat{x})$ is a base for $\prod X_i$.

Answer 1

1.

Sure, that's fine. Your use of "$n$-dimensional" suggests you're only thinking of products of finitely many spaces, which will hold you back and/or lead to confusion when you move on to considering products of infinitely many spaces.

2.

By "generate," surely we mean "take arbitrary unions of,"

Unfortunately, there are lots of meanings of "generate" that you need to use context to distinguish. In this case, "arbitrary unions of" is not the right meaning.

Here, since we have functions rather than open sets, the intended meaning is related to making the projection functions continuous. By thinking through what interpretation might be reasonable (since the discrete topology would easily make the projections continuous), the intended meaning is actually that:

The product topology is the coarsest topology where all the projection functions are continuous. (i.e. the intersection of all topologies that make the projections continuous)