Consider an uncountable $K_\sigma$ set $X\subseteq \omega^\omega$, where $K_\sigma$ means countable union of compact sets and $\omega^\omega$ is the Baire space. Suppose now that there exists a subset $A\subseteq X$ which is dense in $X$ (wrt the subspace topology).
My question is:

• Can we prove that there exists a Cantor set $\mathcal{C}\subseteq X$ such that $\mathcal{C}\cap A$ is dense in $\mathcal{C}$?

Since $X$ is uncountable it must contain a Cantor set, as all analytic sets have the perfect set property, but here we are looking for a Cantor set that preserves the property of $A$ of being dense.

Ideas?
Thanks!

Consider an uncountable perfect $K_\sigma$ set $X\subseteq \omega^\omega$, where $K_\sigma$ means countable union of compact sets, perfect means that $X$ has no isolated points and $\omega^\omega$ is the Baire space. Suppose now that there exists a subset $A\subseteq X$ which is dense in $X$ (wrt the subspace topology).
My question is:

• Can we prove that there exists a Cantor set $\mathcal{C}\subseteq X$ such that $\mathcal{C}\cap A$ is dense in $\mathcal{C}$?

Since $X$ is uncountable it must contain a Cantor set, as all analytic sets have the perfect set property, but here we are looking for a Cantor set that preserves the property of $A$ of being dense.

Ideas?
Thanks!