This is a question with respect to standard topology:
Prove it's false that: the connected components of $K \cup \{0\}$ are singletons.
$$K := \left\{\dfrac{1}{n} \big\vert \, n \in N_{\geq 0}\right\}$$
I already proved that connected components of $K$ are singletons by saying $N$ is a discrete set, thus $K$ is a discrete set, so the only connected open set for $K$ is singletons.
And for this question, I followed the above method and wants to say: we can always find an open set that contains $0$, maybe $(-1,1)$? But not sure how to go from here.
Any help is greatly appreciated!
Let $X=K\cup\{0\}$. Any subset $A$ of $X$ containing two distinct points is disconnected.
Indeed, if we have $1/m,1/n\in A$, with $m>n$ then the set $A$ is the disjoint union of $$ \bigl[(-\infty,r)\cap A\bigr]\cup\bigl[(r,\infty)\cap A\bigr] $$ where $r$ is an irrational number such that $1/m<r<1/n$. The two sets are open in $A$ and not empty.
Similarly if we have $0$ and $1/n$, just take an irrational $r$ with $0<r<1/n$.
Therefore the statement you're assigned with is wrong: the connected components of $K\cup\{0\}$ are singletons is the correct statement.