---- > [!definition] Definition. ([[semilocally simply connected]]) > A [[topological space]] $X$ is called **semilocally simply connected** if any point $x \in X$ has a [[neighborhood]] $U_{x}$ so that any [[parameterized curve|loop]] $\gamma$ in $U_{x}$ based at $x$ is [[homotopy|nulhomotopic]] in $X$ (i.e., $\iota \circ \gamma \simeq e_{x}$ [^1]) > Note that $U_{x}$ doesn't need to be [[simply connected]]: loops just have to be [[homotopy|nulhomotopic]] *in $X$*; the [[homotopy]] need not take place inside $U_{x}$. > [!equivalence] > $X$ is [[semilocally simply connected]] iff any point $x \in X$ has a [[neighborhood]] $U_{x}$ so that the [[group homomorphism|homomorphism]] $\iota_{*}: \pi_{1}(U,x) \to \pi_{1}(X,x)$[[homomorphism of fundamental groups induced by a continuous map|induced]] by [[inclusion map|inclusion]] is trivial. ^equivalence > [!proof]+ Proof of Equivalence. > Unpacking definitions, looks even longer than actually is... $\to$. Let $x \in X$ and $U_{x} \ni x$ such that $\iota_{*}$ is [[group homomorphism|trivial]] ($[\iota \circ \gamma] \equiv [e_{x}]$), i.e., $\iota \circ \gamma \simeq e_{x}$ for any loop $\gamma$ in $U_{x}$. This is saying exactly that $\iota \circ \gamma$ is [[homotopy|nulhomotopic]] in $X$. > $\leftarrow$. Let $x \in X$ and $U_{x} \ni x$ such that any loop $\gamma$ in $U_{x}$ based at $x$ is [[homotopy|nulhomotopic]] in $X$, $\iota \circ \gamma \simeq e_{x}$. So $[\iota \circ \gamma]=[e_{x}]$. ---- #### [^1]: Where $e_{x}:X \to X$ is the constant map $e_{x}(y) \equiv x$ on $X$. ---- #### 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 > ``` #reformatrevisebatch04