Dijkstra's shortest path algorithm without relaxation
This answer refers to a previous version of the pseudocode, which had a bug. Once reaching a vertex, it doesn't update its weight ever again.

Your algorithm isn't Dijkstra's algorithm. Worse, it doesn't calculate shortest paths. As a simple example, consider a graph with nodes $\{s,a,b,t\}$ and the following edges:


$s\to a$ with weight 1.
$s\to b$ with weight 2.
$a\to t$ with weight 9.
$b\to t$ with weight 1.


The shortest path from $s$ to $t$ is via $b$, but your algorithm will discover $t$ via $a$ and will therefore output a distance of 10 rather than the correct distance, which is only 3.

Finally, you mention that the space complexity is $O(E)$. In fact, since the heap and set contain vertices rather than edges, it is $O(V)$.