How to decide if this set is connected or not?

Question

I'm having trouble to know whether the set $E\cup G$ is connected, given that $E,G\subset X$ as well as $X$ are connected sets from the topology $(X,\tau)$ and $\partial E\subset G$. Any help is appreciated. Thanks in advance.

Answer 1

I'm new to but it seem clear that if the boundary of $E$ is a subset of $G$ then $E\cup G$ is connected if $G$ and $E$ are. because all of $\partial E$ is subset to $G$. Hope my comment help.

Answer 2

First, note that $\overline{E} = E\cup\partial E$. Therefore, the assumption that $\partial E\subset G$ means that $\overline{E}\cap G\ne\emptyset$. This implies that your question can be answered by proving the following statement.

Let $X$ be a topological space such that $E,G\subset X$ are connected sets. Assume that $\overline{E}\cap G\ne\emptyset$. Then $E\cup G$ is connected.

This statement has three proofs/hints given by this previous MSE question.