---- > [!definition] Definition. ([[path-connected component]]) > Given a [[topological space]] $X$, define an [[equivalence relation]] on $X$ by setting $x \sim y$ if there exists a [[parameterized curve]] from $x$ to $y$. The [[equivalence class|equivalence classes]] [[partitions are always determined uniquely by equivalence relations|partition]] $X$ into disjoint constituent connected sets, called the **path(-connected) components** or **zeroth homotopy group** of $X$ and collectively denoted $\pi_{0}(X)$. > \ > Each nonempty [[path-connected]] subspace of $X$ intersects exactly one [[path-connected]] component nontrivially. > [!justification] > We need to show this indeed is an [[equivalence relation]]. The identity path witness $x \sim x$. Given a path $\beta:[0,1] \to X$ witnessing $x \sim y$, the 'reverse path' $\gamma(t):=\beta(1-t)$ witness $y \sim x$. Finally, transitivity is shown as follows: let $f:[0,1] \to X$ be a [[parameterized curve]] witnessing $x \sim y$ and $g:[1,2] \to X$ a [[parameterized curve]] witnessing $y \sim z$. Then [[the pasting lemma]] says we can 'paste' $f$ and $g$ together to get a [[parameterized curve]] from $x$ to $z$ (recall that all [[closed interval]]s of $\mathbb{R}$ are [[homeomorphism|homeomorphic]]). > \ > The proof of the second assertion is similar to that seen in [[connected component]]. > [!basicnonexample] Warning. > Despite the name, $\pi_{0}(X)$ is *not* actually a [[group]]! There is not really a 'natural' way to come up with a [[binary operation]] on path components. ^nonexample ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```