Can we really call the environment a public good?

Public goods are defined as good which are non-rival (consuming them doesn't deplete other's ability to consume) and non-excludible (it is not viable to prevent access to this good).

I question whether a good such as the climate, or the air is really non-rival.

Consider air - we can see it having more than 2 uses. One people use to breath, or we can use it to dispose of pollution. Consuming air for breath doesn't change a factories ability to pollute, but a factories pollution does have the cost of air quality for people's breathing.