Unless you have a dirty basement or mold down there than this is totally fine. This is a similar set up to what many folks run.. If you don’t mind the smell you can just run that exhaust unit it comes with outside the tent which filters the air before it goes in.. but then you have nothing to scrub the smell of the grown tent from the amazing bud growing in it!

I have a similar setup. I don't think you'll need an air purifier, but depending on the humidity (or lack) and temperature in the basement you may want to regulate those things so the air going into your room is in the correct ranges. I vent to a small basement window as well, and there's enough open window for fresh air to flow in as well. Though you will eventually want a carbon filter so that you're neighbors don't think you've got a skunk breeding farm in your yard.

Nice! Your plant will appreciate you 😊 I grow in my basement too. It’s a pretty big room, and I don’t have a window. It’s works. I have a exhaust fan and carbon filter inside my tent, and I vent it back into the room. I’ve never had a problem. If you can swing the window- do it. But it’s not a dealbreaker 🌱

For proper c02 without providing it you need fresh air from outdoors. If no outside air then I would use c02 or if you have a water heater that uses gas, you could vent that into your tent instead of outdoors. Anytime you run your hot water then you have some good c02 for your plants to breathe. Cheers