Probably not, or in tiny quantities. All the oxygen we breath, just about, was created through photosynthesis. It is the product of many thousands of years of plants growing - and it is only when plants put on bulk and take carbon out of the atmosphere as wood or peat that the oxygen levels increase.
For instance, in parts of the world with deciduous trees, seasonal changes like leaves growing then falling and decomposing even out with no net increase in oxygen as the oxygen produced in the spring is used up when autumn leaves decay in winter.
To have an increase in oxygen, the peat layers or the layers of soil below a forest have to increase. Or - most of it happens actually in the oceans, where microorganisms produce oxygen and die and fall to the sea bed.
So, for many thousands of years if somehow the sun was blocked, we could breath the air fine. But eventually - all the plants died long before of course - then there would be no new oxygen.
It's different if you live near to a source of powerful ionizing radiation. Then there's another source - the radiation splits water into hydrogen and oxygen. Hydrogen being a light gas may escape if the gravity is low enough. The atmosphere then becomes enriched in oxygen - or it may be that the ice does.
This process may have given early Mars an oxygen rich atmosphere, and also is thought to have made the Europa oceans below the ice oxygen rich.
But the sun is the main source of ionizing radiation for Earth. And shielded by our magnetic fields, then it has less effect on us than it did on early Mars.
Anyway without the sun, then there would be no ionizing radiation so no oxygen.
See also If the Sun disappeared completely, what would happen to the Earth and the rest of our solar system?
For instance, in parts of the world with deciduous trees, seasonal changes like leaves growing then falling and decomposing even out with no net increase in oxygen as the oxygen produced in the spring is used up when autumn leaves decay in winter.
To have an increase in oxygen, the peat layers or the layers of soil below a forest have to increase. Or - most of it happens actually in the oceans, where microorganisms produce oxygen and die and fall to the sea bed.
So, for many thousands of years if somehow the sun was blocked, we could breath the air fine. But eventually - all the plants died long before of course - then there would be no new oxygen.
It's different if you live near to a source of powerful ionizing radiation. Then there's another source - the radiation splits water into hydrogen and oxygen. Hydrogen being a light gas may escape if the gravity is low enough. The atmosphere then becomes enriched in oxygen - or it may be that the ice does.
This process may have given early Mars an oxygen rich atmosphere, and also is thought to have made the Europa oceans below the ice oxygen rich.
But the sun is the main source of ionizing radiation for Earth. And shielded by our magnetic fields, then it has less effect on us than it did on early Mars.
Anyway without the sun, then there would be no ionizing radiation so no oxygen.
See also If the Sun disappeared completely, what would happen to the Earth and the rest of our solar system?
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
4.8m answer views110.3k this month
Top Writer2017, 2016, and 2015
Published WriterHuffPost, Slate, and 4 moreHuffPost, Slate, Forbes, Newsweek, Business Insider, and Science 2.0