# Connected Set and Boundary Problem [closed]

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.

## closed as off-topic by user21820, user99914, GNUSupporter 8964民主女神 地下教會, Shailesh, DRFMay 18 '18 at 14:09

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user21820, Community, GNUSupporter 8964民主女神 地下教會, Shailesh, DRF
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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 $E\cap \partial Y\ne\varnothing$.