Example of product topology where the index set is uncountable

Question

I'm reading through Munkres, chapter 2, section 19 (Product topology). I can't see any example of product topology where the family of space is indexed with an uncountable set.

Can you provide an example which is maybe easy to understand, maybe with some analogy with $\mathbb{R}^n$.

Answer 1

I guess we can do the following (I hope this fits your expectation):

for any $r \in \mathbb{R}$ define the set $X_r = \{x_r, y_r\}$ (two different points).

Define $X = \prod_{r \in \mathbb{R}}X_r$.

If we give each $X_r$ the discrete topology (and so it becomes Hausdorff), then $X$ is a topological space with either the box topology, or the product topology.

An interesting fact about this construction is that in the box topology, this $X$, given that each $X_r$ is Hausdorff, is not compact.