Let X be a locally convex vector space and let G be an open connected subnet of X.

How to show that G is arcwise-connected?

I only can show that G is path-connected but do not know why G is arcwise-connected.

Any help would be appreciated!

  • $\begingroup$ How have you shown $G$ is path-connected? $\endgroup$ – jdc Apr 7 '16 at 4:23
  • $\begingroup$ Is $X$ locally arcwise-connected? $\endgroup$ – Gustavo Apr 7 '16 at 5:16
  • $\begingroup$ @Gustavo.This is an exercise in Conway's <Functional analysis>. I think we can get X is locally arcwise-connected since every $x\in X$ has a neighbourhood $x+V$ in which $V$ is an open balanced neighbourhood of 0. $\endgroup$ – David Lee Apr 7 '16 at 6:23
  • $\begingroup$ @jdc.I only need to show that every path-connected component W of G is open subset. and this can be proved by noticing that for every $x\in W$, there is an balanced open neighbourhood $V_x$of 0 such that $x+V_x \subset$ G. and every point of $x+V_x $ can be connected with x by a path. then $x+V_x \subset W$ and W is open. $\endgroup$ – David Lee Apr 7 '16 at 6:28

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