Can air be pumped into the container completely filled filled with water?
Only to a very slight extent, air can be dissoved in water (think of what fish breathe), but it is a very nonsignificant amount.
No, air cannot be pumped into a container that is completely filled with water. This is because a container that is completely filled with water has no space for any additional substance, including air. If you were to try to pump air into such a container, the air would have no place to go and would simply be unable to enter the container.
To illustrate this, you can think of the container as being already filled to its maximum capacity with water. Just like if you try to pour water into a glass that is already full, the water will overflow, similarly, if you try to pump air into a container that is already filled with water, the air would have nowhere to go and would not be able to enter the container.
To have air inside a container that is filled with water, there needs to be some air space or an air pocket present in the container initially, before filling it with water. In such a case, when the container is filled with water, the air pocket gets trapped inside the container along with the water, resulting in a mixture of air and water within the container.