Prim's algorithm

Algorithm.
1. Start at any vertex $r$. Set $T=\{r\}$, priority queue of edges incident to $T$ with weights.
2. Repeatedly: pick the minimum-weight edge from $T$ to $V\setminus T$, add its other endpoint to $T$ and the edge to the MST.
3. Stop when $T=V$.

Complexity. $O(|E|\log|V|)$ with a binary heap.

Conclusion. Prim computes an MST.