Connected and path components

Question

• Components are always closed in the topology of the whole space (why?) but open in the subspace topology (of what?)?
• Need to justify the claim that $C(x)$ is the largest connected set containing $x$?

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Overview §

Definition: (Connected) components

Given any points $x,y$ in a topological space $X$, define an equivalence relation by $x{\sim }_{C}y$ if there exists a connected subspace $A\subseteq X$ such that $x,y\in A$. The components of $X$ are the equivalence classes under ${\sim }_C$, and the component of a point $x\in X$ is denoted

$$C(x)=\{y\in X|y{\sim }_{C}x\}.$$

Definition: Path components

Given any points $x,y$ in a topological space $X$, define an equivalence relation by $x{\sim }_{P}y$ if there exists a path from $x\to y$. The path components of $X$ are the equivalence classes under ${\sim }_P$, and the path component of a point $x\in X$ is denoted

$$P(x)=\bigcup _{\begin{array}{c}x\in A\subseteq X\\ A\text{ is path connected}\end{array}}A.$$

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Examples §

Components of $\mathbb{Q}$ are singletons §

If $x\in \mathbb{Q}$, then $C(x)=\{x\}$. Otherwise, if $x\in A\subseteq \mathbb{Q}$, and there is some $y\in A$ such that (without loss of generality) $x<y$, then $x<z<y$ for some irrational $z\in \mathbb{R}$. Then $A\cap (-\infty ,z)$ and $A\cap (z,\infty )$ are a separation of $A$—this is similar to showing that $\mathbb{Q}$ is not connected.

Topologist’s sine curve §

Topologist’s sine curve has one component but two path components.

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Review §

• Check that connected components and path components are indeed equivalence relations (reflexive, symmetric, transitive).
• Is a connected component $C(x)$ open or closed?
• Are any components of $\mathbb{Q}$ open?
• Check that the path component $P(x)$ is indeed the union of all path-connected subsets $A\subseteq X$ that contain $x$. ⭐