0

Let $X$ be a topological space and $a \in X$. Let $\Gamma_a$ be the collection of all clopen sets in $X$ that contain $a$. Let $C$ be the connected component of $X$ that contains $a$.

I'm trying to prove a result mentioned in this thread. Could you have a check on my attempt?

Theorem: If $A \in \Gamma_a$ then $C \subset A$. This implies $C \subset\bigcap \Gamma_a$.

Proof: Let $\Lambda_a$ be the collection of all connected subsets of $X$ that contain $a$. Then $C = \bigcup \Lambda_a$. Fix $A \in \Gamma_a$ and $B \in \Lambda_a$. Then $\{B \cap A, B \cap A^c\}$ is a partition of $B$ where $B \cap A$ and $B \cap A^c$ are both clopen in $B$. Then either $B \cap A = \emptyset$ or $B \cap A^c = \emptyset$. Because $a \in A \cap B$. Then $B \subset A$. This completes the proof.

Akira
  • 17,367

1 Answers1

1

Your proof is correct, but it can be simplified. You do not need the collection $\Lambda_a$. Simply take $C$ (which is by the way, also a member of $\Lambda_a$) instead of an arbitrary $B \in \Lambda_a$.

The intersection of all clopen sets in $X$ that contain $a$ is known as the quasi-component of $a$. You proved the well-known theorem that the component of a point $a \in X$ is always contained in its quasi-component.

Quasi-components and components coincide for compact Hausdorff spaces.

Kritiker der Elche
  • 4,417