Are every maximal connected(resp. path connected) sets, components(resp. path components)?

Question

I know, every components (resp. path components) are maximal connected (resp. path connected) sets. But is the converse true?

Answer 1

I assume you define (path) components as follows:

Let $X$ be a topological space. Define a relation: $x\sim y$ iff there exists a (path) connected subset $A\subseteq X$ such that $x,y\in A$. A (path) component is an equivalence class of $\sim$.

With that we have

A subset $A\subseteq X$ is a (path) component if and only if $A$ is a maximal (path) connected subset.

Proof. "$\Rightarrow$" Assume that $A$ is a (path) component and $A\subseteq B$ for some (path) connected $B$. Then by the definition $x\sim y$ for any $x, y\in B$, in particular for $x\in A$. Since $A=[x]_\sim$ then it follows that $B\subseteq A$ and thus $A$ is maximal.

"$\Leftarrow$" Assume that $A\subseteq X$ is a maximal (path) connected subset. Since it is (path) connected then any $x,y\in A$ are in relation. Thus $A\subseteq [x]_\sim$. Assume that there is some $y\in X-A$ such that $x\sim y$. Then $x,y\subseteq C$ for some (path) connected subset $C$. It is easy to see that $A\cup C$ has to be (path) connected as well (since they intersect nonempty) and thus $A$ is not maximal. Contradiction. $\Box$