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Prove that any open connected set in $C[0,1]$ is polygonally connected.

(Here $C[0,1]$ is the space of real valued continuous functions on $[0,1]$ with the metric: $d(f,g)=$ $sup${$|f(x)-g(x)| : 0\leq x\leq 1$}; a polygonal path is a path made up of a finite number of line segments)

WhySee
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1 Answers1

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The proof is pretty much the same as, say, the proof that an open connected set in a locally path-connected space is path-connected. It does not matter that you are considering $C[0,1]$ rather than any other normed vector space:

Let $(V, \|\cdot\|)$ be a normed vector space (or more generally, let $V$ be a locally convex topological vector space).

  1. Show that any open ball in $V$ is polygonally connected.

  2. Let $U$ be a non-empty connected open set in $V$. Let $x_0 \in U$ and define $A \subset U$ as the sets of points $x \in U$ such that there exists a polygonal path in $U$ joining $x_0$ to $x$. Show that $A$ is open and closed in $U$ (use 1.) and conclude.

Seub
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