Goldfish are hardy creatures that can thrive without an air pump, yet still require water aeration and filtration to remain healthy. Just as humans prefer clean air over smog-filled air, goldfish don't like swimming in murky water; therefore an aquarium equipped with sufficient surface disturbance and oxygenating bubbles as well as regular water changes is key in keeping its tank clean and healthy for its inhabitants.
Due to its small size, a fishbowl may not provide an ideal environment for goldfish; waste will build up quickly, leading to oxygen deprivation in the water. Therefore, it is recommended that aquatic plants are added for oxygenation purposes and that you regularly change out your water in order to prevent waste build-up.
An aquarium tank is essential in providing your goldfish with optimal conditions, and using an air pump to provide adequate aeration and bubbles will keep their health optimal. Furthermore, adding an air pump helps regulate water temperature to stay within acceptable limits and maintain proper ranges for temperature regulation.
Watch your goldfish carefully to assess whether they're receiving enough oxygen. If they seem distressed and seem to gulp for air near the surface of the water, this indicates their oxygen levels have dropped dangerously low and need immediate action taken to rectify it. Another telltale sign would be rapid movement of their gills indicating struggled oxygen intake through their gills; this should also prompt immediate intervention from an aquarium professional.