I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are true or not?
Statement 1
Let $X$ be a $T_1$ space. Let $C$ be a connected subset of $X$. If $C$ is finite, then $|C|≦1$.
Statement 2
Let $X$ be a metric space. Let $C$ be an infinite connected subset of $X$. Then, $\forall p\in C$, $p$ is a limit point of $C$.
Both are correct.
For the first, if you assume that it is not a singleton or the empty set, you can use the "Hausdorffness" to create for every $x$ of $C$ an open set that separates $x$ from the rest of $C$ because $C$ is finite. By this you create a finite separation of $C$, and this is a contradiction to its contentedness.
For the second, every point in a set $A$ is a limit point of $A$ always, since you can take the constant sequence. But I guess you meant that for every $x$ in $C$, $x$ is a limit point of $C-\{x\}$. In that case you can assume that it is not so. Then there's a subset of $X$, $U$, an open neighborhood of $x$ so that does not intersect with $C-\{x\}$ and let's say that $V$ is the intersection of $U$ and $C$, it's open in $C$, and $C-\{x\}$ is also open in $C$ since $\{x\}$ is closed in $X$ (metric space), so $V$ and $C-\{x\}$ form a separation of $C$, hence $C$ is not connected.