Let $E$ be a real elliptic curve with identity element $O \in E(\mathbb{R})$, and endow the real locus $E(\mathbb{R})$ with its usual Euclidean topology. If $E(\mathbb{R})^0$ denotes the connected component of $E(\mathbb{R})$ containing $O$, then $E(\mathbb{R})^0$ is a subgroup of the topological group $E(\mathbb{R})$.