Uniform closures are uniformly closed but pointwise closures might not be pointwise closed

Question

An old set of classroom notes I took in a math course says the following:

For a set ${X}$ let $\mathcal{A}$ be a set of functions from $X$ to $\mathbb{R}$. Then

1. the uniform closure of $\mathcal{A}$ is uniformly closed.
2. the pointwise closure of $\mathcal{A}$ is not necessarily pointwise closed.

The prof said the first assertion was straightforward (we proved it as homework) but that the second was not (I came up with a counter-example).

I'd like to write up the counter-example in a short note, but I don't even know what field of math this is in. Topology? Real analysis? Specifically, I'd like some easily accessed (preferably online) citations that discuss and give background to these assertions. So far my online searches have landed on discussions that assume the functions form an algebra, or have restrictions such as boundedness or continuity.

Answer 1

A suitable context for discussion of the assertions is the concept of Baire functions and Baire classes. Quoting from the linked article:

Baire functions of class α, for any countable ordinal number α, form a vector space of real-valued functions defined on a topological space, as follows.

• The Baire class 0 functions are the continuous functions.
• The Baire class 1 functions are those functions which are the pointwise limit of a sequence of Baire class 0 functions.
• In general, the Baire class α functions are all functions which are the pointwise limit of a sequence of functions of Baire class less
  than α.

So, if $\mathcal{A}$ is the set of continuous functions, its pw(pointwise) closure is the set of Baire class 1 functions, and the pw closure of that is the set of Baire class 2 functions, etc. In other words, the pw closure of $\mathcal{A}$ is not pw closed.

Coming up with more restricted examples of $\mathcal{A}$ is another problem.

Answer 2

This is topology within real analysis and covered by, for example, "Baby Rudin" Principles of Mathematical Analysis, Walter Rudin, page 144 in the 3rd edition. You also need the concept of a Banach space with the "point-wise" max norm. See the table of classical Banach spaces, specifically C(X) here: https://en.wikipedia.org/wiki/Banach_space and here https://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space