Leave $ G $ be a topological group and $ H $ A closed subgroup. Define $ pi: G a G / H $ by $ g mapsto gH $ and equip $ G / H $ with the topology induced by $ pi $, that is, the best topology such that $ pi $ It is continuous A text that I am reading affirms that it is "clear" that $ pi $ It is an open map, that is, for each open $ V in G $ we have that $ pi (V) $ It's open.

This is not clear to me at all.

A necessary and sufficient condition for $ pi (V) $ be open in $ G / H $ is for the whole

$$ bigcup_ {v in V} vH $$

be open in $ G $. I don't know how I can say anything like this since this may be a larger set than $ V $, and who knows which sets are or are not in the topology of $ G $.

I suspect that I'm missing something very obvious?