Is there anything in the Bolzano-Weierstrass proof (for points in $\mathbb{R}^k$) that is analogous to using equicontinuity for functions in Arzela-Ascoli? Does the statement: "points are always equicontinuous" make sense? I feel that I may be forcing this reinterpretation, but I also feel that there might be some connection between the topology of points and the topology of a family of equicontinuous functions?
This question originiated from a previous discussion in which one nice interpretation was offered:
perhaps we can think of points as (trivially equicontinuous) constant functions
Yes, a point in $\mathbb R^k$ can be viewed as an element of $C(K)$, where $K$ is the compact set $ \{1, 2, \dots, k \}$. Since $\{1, 2, \dots, k \}$ is a finite union of discrete points, every function $\{1, 2, \dots, k \} \to \mathbb R$ is automatically continuous, so $C(K)$ does indeed correspond to the whole of $\mathbb R^k$. Furthermore, every family of functions $\{1, 2, \dots, k \} \to \mathbb R$ is automatically equicontinuous, so Arzela-Ascoli reduces to the statement that a subset of $C(K)$ has compact closure iff it is bounded.
[Note that if we think of $\mathbb R^k$ as $C(K)$ with $K = \{1,2,\dots,k \}$, then the natural norm on $\mathbb R^k$ would be the $l^\infty$ norm, rather than the Euclidean $l^2$ norm. However, all norms on a finite-dimensional vector space induce equivalent topologies, so this is not a problem.]