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Given a first category set $A \in [0,1]$, is the set $X = \{f \in C[0,1]:f(A)\text{ is first category}\}$ residual in $C[0,1]$?

I tried two strategies: First I tried to play the Banach-Mazur game on $X^c$, but it does not seem clear to me who would have a winning strategy. Secondly, I tried finding a dense $G_\delta$ set contained in X, but it did not get me very far either. How do I approach this problem?

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  • $\begingroup$ What is the Banach-Mazur game? What do you mean by "residual" -- a set of first category? $\endgroup$ – uniquesolution Apr 29 at 9:04
  • $\begingroup$ @uniquesolution Given a complete metric space $X$, a set $A \subset X$ and two players, player 1 wins if $(\bigcap^\infty_{i = 1} I_i) \cap A \neq \emptyset $ and player 2 wins if the intersection is empty. Players have to choose $I_i$ such that they are nested nondegenerate closed balls. It can be shown that player 2 has a winning strategy iff $A$ is first category. Residual means that its complement is first-category. $\endgroup$ – Ustad Kadir Misiroglu Apr 29 at 9:20
  • $\begingroup$ Which topology do you consider on $C[0,1]$? $\endgroup$ – Alex Ravsky Jun 18 at 3:36
  • $\begingroup$ @AlexRavsky the topology induced by the sup norm: $\| f(x)\| = sup_{0 \leq x \leq 1} |f(x)|$ $\endgroup$ – Ustad Kadir Misiroglu Jun 28 at 6:56

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