Let $X$ be a topologic space. Let $U,V\subseteq X$ be open, disjoint such that $U\cup V=X$ and let $A\subseteq X$ be connected. Then $A\subseteq U$ or $A\subseteq V$
Let assume that $A\not\subseteq U$ or $A\not\subseteq V$, because $U\cup V=X$ we get $A\cap V\neq \emptyset$ and $A\cap U\neq \emptyset$ corresponding. $A\cap V$ and $A\cap U$ are open, non empty, disjoint sets such that: $$(A\cap U) \cup (A\cap V)=A\cap(U\cup V)=A\cap X = A$$
Therefore $A$ is not connected. Contradiction
Why can we assume that $A\cap V$ and $A\cap U$ are open? I understand that $U,V$ are open, but why $A$ is open?