Stop me if you've heard this one: A fan, a heater, and an air purifier all walk into a bar. No wait, a Dyson. Sorry. Also, there's no punchline.

For several years, Dyson has sold a stylish, bladeless fan. For slightly fewer years, it has sold a fan that also heats. For a few months, it has sold a fan that also purifies the air. On September 6, the innovative engineering company's Pure Hot+Cool Link will complete the seemingly inevitable Venn diagram. It heats, it cools, it purifies the air. It's a multitool for in-home comfort.

Having to Frankenstein together three features into one sleek cylindrical cyborg took a bit of work, though not in the way one might expect. Fitting all the pieces together? Not so tough.

A connected app will register air quality history so you can track which days are particularly sneeze-inducing.

"When you look at our desk purifier, you see a little bit of similarity [to the Pure Hot+Cool Link] from the filter down. They're about the same size," says Dyson design engineer Sean Hopkins. "And then when you look from the filter up, you see essentially everything in the A09," Dyson's cooling/heating fan. Integrating those was relatively simple, next to the broader challenge of making heating and air purification play nice together.

The problem, Hopkins explains, is that heating a room can affect its air quality. Heat and humidity can exacerbate allergens, which an air purify has to work harder to clear out.

"It comes down to taking into account how do we make the most efficient use of the purifier function while maintaining the room at a certain temperature," says Hopkins. "It's a well-timed balancing act, or more like a see-saw."

Apparently navigating those trade-offs haven't affected the Pure Hot+Cool Link's effectiveness; Dyson claims it will remove 99.97 percent of particles—think pollen, mold, air pollution—in its given location, all while keeping hot days cool and cold days comfortable. A connected app will also register air quality history, so that you can track which days are particularly sneeze-inducing. It can also let you control just how clean you want your air to be, if you'd rather save your energy bill more than your lungs.

It's a $600 device, which at first sounds like a lot, but less so when you remember that it's a three-for-one special.