self-intersections of a curve

self-intersections of a curve

Let $X$ be a topological manifold and $\gamma :[0,1]\to X$ a segment of a curve in $X$.

Then the curve is said to have a self-intersection in a point $p\in X$ if $\gamma$ fails to be injective, i.e. if there exists $a,b\in (0,1)$, with $a\ne b$ such that $\gamma (a)=\gamma (b)$.Usually, the case when the curve is closed i.e. $\gamma (0)=\gamma (1)$, is not considered as a self-intersecting curve.