Connected Set and Boundary Problem

Question

Let $(X,\tau)$ be a topological space and $E,Y\subset X$ such that $E$ is connected, $E\cap Y\neq\emptyset$ and $E\cap(X\setminus Y) \neq \emptyset$. Thus $E\cap\partial Y \neq \emptyset$. Any help is appreciated. Thanks.

Answer 1

Recall that $X$ is the disjoint union of $Y^\circ$, $\partial Y$, and $(X\setminus Y)^\circ$. Suppose $E\cap\partial Y=\varnothing$, Then $E$ is the disjoint union of $E\cap Y^\circ$ and $E\cap(X\setminus Y)^\circ$. But this is a separation of $E$ (in the subspace topology), a contradiction. It follows that $E\cap \partial Y\ne\varnothing$.