Your compactum $X$ is finitely Suslinian (by defnition). As such, every connected subset of $X$ is locally connected (see this paper), and this implies that the components of $X\setminus C$ are the same as the quasi-components of $X\setminus C$. See this paper, Theorem 2.1. @Fedor Petrov already argued that $x$ and $y$ are in different components of $X\setminus C$, therefore they are also in different quasi-components.

Your compactum $X$ is finitely Suslinian (by defnition). As such, every connected subset of $X$ is locally connected (see this paper), and this implies that the components of $X\setminus C$ are the same as the quasi-components of $X\setminus C$. See this paper, Theorem 2.1. @Fedor Petrov already argued that $x$ and $y$ are in different components of $X\setminus C$, therefore they are also in different quasi-components.

Note that $C$ does not have to be connected or compact. It can be any subset of $P$ such that $P\setminus C$ is disconnected.

Your compactum $X$ is finitely Suslinian (by defnition). As such, every connected subset of $X$ is locally connected (see this paper), and this implies that the components of $X\setminus C$ are the same as the quasi-components of $X\setminus C$. See this paper. @Fedor Petrov already argued that $x$ and $y$ are in different components of $X\setminus C$, therefore they are also in different quasi-components.

Your compactum $X$ is finitely Suslinian (by defnition). As such, every connected subset of $X$ is locally connected (see this paper), and this implies that the components of $X\setminus C$ are the same as the quasi-components of $X\setminus C$. See this paper, Theorem 2.1. @Fedor Petrov already argued that $x$ and $y$ are in different components of $X\setminus C$, therefore they are also in different quasi-components.