Clopen subsets in topological subspaces I have this problem and am not sure where to start:
Say $$A = \{(0,0),(0,1)\} \cup \bigcup^∞_{n=1}  S_n$$ is a subset of the euclidean plane that inherits the standard topology from said plane.
Here $S_n = \{(\frac1n, y) : 0 \le y \le 1\}$ for all $n\ge1$. Then let $B$ be a non empty subset of $A$, and $B$ is clopen in $A$.
The objective is to show that if $B$ contains either $(0,0)$ or $(0,1)$, then it contains both. I don't understand where to go or where to start here. Constructing a set where the complement is open is proving tough, and even then I don't see how that relates to anything.
Any advice?
 A: In general, a clopen set cannot intersect a connected set partially. That is, if $B$ is clopen and $S$ is connected then $U:=B\cap S\ne\emptyset$ and $V:=B^c\cap S\ne\emptyset$ is impossible. Otherwise, $U,V$ are open sets in $S$ (by definition of open set in the subspace $S$), they are non-empty, they are disjoint, and their union is $S$; by definition this would make $S$ disconnected. (In particular, if $B\cap S\ne\emptyset$ then $V=\emptyset$ and so $S\subseteq B$.)
In the question, we are given a clopen set $B$ in $A$. Each line $S_n$ is connected (in the standard topology), so $B$ cannot partially intersect it, by the above. For each $n$, either $S_n\subseteq B$ or $S_n\cap B=\emptyset$.
Now suppose $(0,0)\in B$. Since $B$ is open, $(0,0)$ is an interior point of it, meaning that there is a small ball $C$ centered at $(0,0)$ such that $C\cap A\subseteq B$. But any such ball intersects all $S_n$ for $n\ge N$ (for some $N$). Thus $S_n\subseteq B$ for $n\ge N$. Hence, $(\frac{1}{n},1)\in S_{n}$ is a sequence in $B$ (at least for $n\ge N$), converging to $(0,1)$. As $B$ is closed, $(0,1)\in B$.
