Can someone give me an example of the following:
Let $X$ be a topological space and $x_1, x_2 \in X$. Suppose that for every separation $X= A \cup B$, the points $x_1, x_2$ are both in $A$ or both in $B$. Give an example such that $x_1$ and $x_2$ are not in the same (connected) component of X.
Put $X=\{ (0,0),(1,0)\cup \bigcup_{n\neq 1}\{ (u,1/n)|u\in\mathbb{R}\}\},\ x=(0,0),\ y=(1,0)$.
If there are two disjoint open not empty $A_{x},A_{y}\subseteq X$ such that $A_{x}\cup A_{y}=X$ then we can take, without lost of generality, $z\in\{(u,1/m):z\in\mathbb{R}\}\cap A_{x}$, for some $m\in\mathbb{N}$. Then $\{(u,1/m):z\in\mathbb{R}\}\cap A_{y}=\emptyset$, because if not, $\{(u,1/m):z\in\mathbb{R}\}\cap A_{x}$ and $\{(u,1/m):z\in\mathbb{R}\}\cap A_{y}$ is "disconnection" of $\{(u,1/m):z\in\mathbb{R}\}$. But $\{y\}$ is not open in $X$. $\rightarrow\leftarrow$
Since every $\{ (x,1/n)|x\in\mathbb{R}\}$ is a connected component of $X$, we deduce that $x$ and $y$ are in the same component.