For $x\in X$ Prove $C(x)$ is a connected maximal in $X$

Question

For $x\in X$ Prove $C(x)$ is a connected maximal set in $X$

I don't understand the definition of connected maximal set very clear and i don't know how start this exercise, can someone help me with a hint?

Answer 1

If maximality is not the definition, then probably the definition is

$$C(x) = \bigcup \{ C \subseteq X: x \in C, C \text{ connected}\}$$

This is a connected set (which contains $x$), as a union of connected sets that all intersect (in $x$). The union is non-void as $C= \{x\}$ is always part of that union.

And if $C$ is any connected set that contains $x$ then it is by definition one of the sets in the union that comprises $C(x)$ and so trivially $C \subseteq C(x)$, and so if $D$ is connected and $C(x) \subseteq D$, we know $x \in D$ so $D \subseteq C(x)$ by the above and hence $C(x) = D$.

($m$ is a maximal element in a partially ordered set $(P, \le)$ iff

$$\forall p \in P: p \ge m \implies p=m$$ and we have shown that a component is a maximal element in the set of all connected subsets of $X$ ordered by $ \subseteq$.)