Can inhaling the dust of cat poop cause toxoplasmosis?

Swallow, not breathe. You need to swallow toxoplasmosis in order to get sick. So if cat poop dust gets on your hands and you forget to wash your hands, you could possibly transfer the toxoplasmosis from your hands to your mouth. You can't catch it from breathing.

Related Questions

I'm pregnant. My dog rolled in cat feces and then scratched himself causing hair to fly everywhere. Can I catch toxoplasmosis by inhaling hair/dust?

Extremely unlikely. That sounds like an incredibly unlikely wait to catch toxoplasmosis. Simply wash your hands and it won't be an issue. Toxoplasmosis is transmitted by ingestion not inhalation.