What’s The Deal With Hausdorff Spaces? (Part II)

Last time, we learned that—in a world in which much is uncertain—at least we can trust continuous maps of Hausdorff spaces to behave nicely with respect to dense subsets.

Or can we? We showed that if $S\subset X$ is dense, then any continuous map $f: X\to Y$ is determined by its values on $S$, which jives well with our intuition about maps $f :\mathbb{R}\to\mathbb{R}$, for example.

But in $\mathbb{R}$ we can go ever farther. For any open set $U\subset\mathbb{R}$, we can construct a bump function that is nonzero on $U$, but zero outside of $U$. It follows that if $S\in\mathbb{R}$ is not dense, then continuous functions $f:\mathbb{R}\to\mathbb{R}$ are not determined by their values on $S$.

Is this true of all Hausdorff spaces?

The answer is yes, but proving it requires some creativity. The brute force approach does not work here—there is no clear way to create a bump function on some arbitrary space. If you consider yourself a point-set topology guru, I encourage you to try to prove the proposition yourself before reading on.

Proposition: If $X$ is a Hausdorff space and $S\subset X$ is not dense, then there exists some Hausdorff space $Y$ and two distinct continuous functions $f,g: X\to Y$ that agree on $S$.

Proof: Let $K=\overline{S}$, and let $Y$ be two copies of $X$ glued together along $K$. Since there are points not in $K$, the two embeddings $f,g:X\to Y$ are distinct, but agree on $K$ by construction.

I have kept the proof (extremely) short to illustrate the beauty of the idea, but there are a few assumptions that need to be justified. The most serious of these is the assertion that $Y$ is Hausdorff. But a little reading on disjoint unions and quotient spaces will reveal that this is not particularly difficult. (though it is very necessary to know that $K$ is closed)

I do not know if this result is as strong as possible. So I leave the question to you, if you hunger and thirst for more than these two articles have provided.

Question: Does there exist some $T_1$ space $X$ and a non-dense subset $S\subset X$ such that any continuous map $f:X\to Y$, where $Y$ is Hausdorff, is determined by its values on $S$?