$X$ be a topological space , then is it true that every connected component of $X$ is a union of path-connected components?

Question

Let $X$ be a topological space , then is it true that every connected component of $X$ is a union of path-connected components ? I only know that for every point , its path connected component is contained in its connected component . Please help . Thanks in advance

Answer 1

For a point $p \in X$ let $C_p$ be the path-connected component. Moreover let $C$ be a connected component of $X$. Then $$C = \bigcup_{p \in C} C_p.$$ This directly follows from $p \in C_p$, thus we have $\subseteq$, and your comment that the path-connected component is contained in its connected component, hence $C_p \subseteq C$ for all $p \in C$ and therefore $\supseteq$.