All the countably infinitely many guests of Hilbert's Hotel decide to spend the day playing tag in the park. One player is the runner, and all the others are it. The taggers can agree on a strategy beforehand, and can communicate and coordinate instantanteously and continuously while the game is played. The runner runs at speed $v_r$, the other players all run at speed $v_{\text{it}}$. The players are allowed to make sharp turns (i.e., their trajectory is not necessarily continuously differentiable); they may even engage in (non-random) Brownian motion.
Question. Can the runner avoid the others players indefinitely, if the park is a connected open set $P\subseteq \mathbb R^2$, with finite area and circumference?
I suspect that it is necessary that $v_r>v_{\text{it}}$, because otherwise a single player can catch up with the runner. (To be precise, if $v_r<v_{\text{it}}$ then a player will simply gradually catch up with the runner by running straight towards her; otherwise, if $v_r=v_{\text{it}}$, we use the fact that the park has finite circumference, so the runner will have to make sharp turns infinitely many times, and this is when the player gains on the runner).
Some observations:
- The runner loses to uncountably many taggers, who can close in on her by forming a closed loop
- The runner wins against one tagger, around whom she runs circles
- The runner wins against finitely many taggers: she "dodges" her adversaries one by one with maneuvers which she can make arbitrarily small
As you said if $v_r < v_{it}$ the its can trivially catch the runner in finite time, as you already said. But then the situation is somewhat problematic. Consider this: Having countably many catchers means that it is possible to place a catcher onto each spot of a countable dense subset of the area.
Arguably now the runner would have a hard time escaping, because no matter where he moves, there is already a catcher arbitrarily much closer to him. But on the other hand he cannot get caught, as no matter where the runners are placed there will always be plenty more points arbitrarily close than the runners could populate.
No need to speak, in such a situation speed does not really make any sense anymore. And the answer depends on you model.