Path-Connected Components of Locally Path-Connected Spaces are Open
# Statement
Let $X$ be a Topological Space that is Locally Path-Connected. Then every Path-Connected Component $P \subset X$ is Open.
# Proof
Let $P \subset X$ be a Path-Connected Component of $X$. Since $X$ is Locally Path-Connected, for each $x \in P$, there exists $U \subset X$ Open so $x \in U$ and $U$ is Path-Connected. Since $U \cap P \ni x$, we know that $U \cup P$ is Path-Connected because If Path-Connected Sets share a point, then their Union is Path-Connected. But $U \cup P \supset P$. Because $P$ is Maximal, $U \cup P = P$. Thus $U \subset P$. Since $x$ was arbitrary, we can find such a $U_{x}$ for each $x \in P$. But then $$P = \bigcup\limits_{x \in P}U_{x}$$so $P$ is Open. $\blacksquare$