## Why is $\P_d$ closed in $C([a,b])$?

Recall that $\P_d$ is the set of all real polynomials of degree at most $d$, and we consider the space $C([a,b])$ equipped with the $\sup$ norm. The claim is that $\P_d$ is a closed subset of $C([a,b])$. This is actually quite tricky to show—at least, I don’t know of a simple argument. Here is what…

## Math 4A03: Proof of the last lemma on connectedness

Let $(M,d)$ be a metric space; we consider $M^2$ with the corresponding product metric. The goal of this post is to give a proof of the last lemma on these slides: Lemma Let $A,B \subseteq M$ be connected. Then $A \times B$ is connected. The proof uses the exercises preceding this lemma given on the…

## Analytic continuation of $\log$-$\exp$-analytic germs

Our preprint on the analytic continuation of germs at $+\infty$ of unary functions definable in $\Ranexp$ is now on the ArXiv. Here is its introduction: The o-minimal structure $\Ranexp$, see van den Dries and Miller or van den Dries, Macintyre and Marker, is one of the most important regarding applications, because it defines all elementary…